Lesson-3 TRIGONOMETRIC RATIOS AND IDENTITIES

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1 Lesson- TRIGONOMETRIC RATIOS AND IDENTITIES Angle in trigonometry In trigonometry, the measure of an angle is the amount of rotation from B the direction of one ray of the angle to the other ray. Angle may be positive or negative and can be of any magnitude. For eample if OA and OB be the positions of revolving rays, the angle formed will be AOB. O A Measurement of Angles The angles are measured in degrees or in radians which are defined as follows : Degree : Radian : A right angle is divided into 90 equal parts and each part is called a degree. Thus a right angle is equal to 90 degrees. One degree is denoted by º. A degree is divided into sity equal parts and each part is called a minute and is denoted by. A minute is divided into sity equal parts and each part is called a second and is denoted by. Thus, we have: right angle 90º º A radian is the angle subtended at the centre of a circle, by an arc equal in length to the radius of the circle. In the figure OA OB arc AB r radius of the circle; the measurement of AOB is one radian and is denoted by c. (Note: usually we write c as ) The ratio of the circumference of the circle to the diameter of the circle is always equal to a constant and this constant is denoted by. Thus, Trigonometric Ratios circumference ; Further c 60 diameter C B r C O r A In a right angled triangle ABC, if BAC then the si trigonometric ratios are defined as follows: sin cos tan Perpendicular Hypotenuse Base Hypotenuse B H P H P B H H, cot, cosec, sec B P P B A H B P C B

2 Signs of Trigonometric Ratios st quadrant : 0 < < 90º, all trigonometric ratios, are +ve. y nd qudarant : 90º < < 0º, only sin and cosec are +ve. rd quadrant : 0º < < 70º, only tan and cot are +ve th quadrant : 70º < < 60º, only cos and sec are +ve nd sin, cosec are +ve rd tan, cot are +ve O st All Ratios are +ve th cos, sec are +ve Limits of the value of trigonometric functions sin cos sec or sec cosec or cosec (e) < tan < (f) < cot < Allied Angles The angles n and n, where n is any integer, are known as allied or related angles. The trigonometric functions of these angles can be epressed as trigonometric functions of, with either plus or minus sign. The following working rules can be used in determining these functions:. Let 0 < < 90. Find the quadrant in which the given allied angle lies. The result has a plus or a minus sign according as the given function is positive or negative in that quadrant.. If n is even, the result contains the same trigonometric function as the given function of the allied angle but if n is odd, the result contains the corresponding co-function i.e. sine becomes cosine, tangent becomes cotangent, secant becomes cosecant and vice-versa.. To determine sin (60º ), we note that 60º 7 90º, which belongs to the third quadrant if 0 < < 90º. In this quadrant sine is negative and since given angle contains an odd multiple of, sine is replaced by cosine sin (60º ) cos. To determine cos (70º ), we note that 70º 90º, is in the th quadrant if 0 < < 90º. In this quadrant, cosine is positive and since given angle contains an even multiple of, cosine function is retained. Hence cos (70º ) cos.

3 Trigonometric Ratios of Compound Angles sin cos tan sin cos tan sin cos tan sin cos tan cos sin cos sin cos sin cot tan cot sin sin cos cos tan tan sin cos tan cos sin cos sin cos sin cot tan cot sin sin sin cos cos cos tan tan Sine, Cosine and Tangent of some angles less than 90 tan Angles 0º 5º º 0º 6º 5º 60º 90º sin cos tan not defined Algebraic sum of two or more angles is called a compound angle. If A, B, C are angles then A + B, A B, A + B + C, A B + C, A B C, A + B C, etc, are all compound angles. Addition and Subtraction Formulae sin(a + B) sina cosb + cosa sin B sin(a B) sina cosb cosa sinb cos(a + B) cosa cosb sina sinb cos(a B) cosa cosb + sina sinb

4 tan(a + B) tan(a B) tan A tan B tan A tan B tan A tan B tan A tan B sin(a + B) sin(a B) sin A sin B cos B cos A cos(a + B) cos(a B) cos A sin B cos B sin A sin A sina cosa tan A tan A cos A cos A sin A sin A cos A tan tan A A tan A tan A tan A sin A sina sin A cos A cos A cosa tan A (tan A tan A) tan A Product Formulae sina cosb sin(a + B) + sin(a B) cosa sinb sin(a + B) sin(a B) cosa cosb cos(a + B) + cos(a B) sina sinb cos(a B) cos(a + B) Formulae on Sums and Differences sinc + sind sin sinc sind cos cosc + cosd cos cosc cosd sin C D C D C D C D cos sin C D C D cos sin C D D C

5 Conditional Trigonometric Idenitities Identities : A trigonometric equation is an identity if it is true for all values of the angle or angles involved. Conditional Identities : When the angles involved satisfy a given relation, the identity is called conditional identity. In proving these identies we require the properties of complementary and supplementary angles Some Important Conditional Identities : If A + B + C, then tana + tanb + tanc tana tanb tanc cota cotb + cotb cotc + cotc cota sina + sinb + sinc sina sinb sinc cosa + cosb + cosc cosa cosb cos C cos A + cos B + cos C cos A cos B cos C cosa + cosb + cosc + sin A sin B sin C tan A tan B + tan B tan C + tan C tan A cot A B C A B C + cot + cot cot cot cot Triple Angle Formulae sin sin(60º ) sin(60º + ) sin cos cos(60º ) cos(60º + ) cos tan tan(60º ) tan(60º + ) tan Maimum and Minimum values of a cos + b sin Consider a point (a, b) on the Cartesian plane. Let its distance from origin be r and the line joining the point and the origin make an angle with the positive direction of -ais. Then, a r cos and b r sin a cos + b sin r (cos cos + sin sin ) r cos ( ) r a cos + b sin r as cos ( ) Hence, the maimum value is a b and minimum value is a b

6 SOLVED EXAMPLES E.: Prove that sec sec cosec + cosec cot tan. Sol.: L.H.S. sec sec cosec + cosec [sec cosec ] + cosec sec [sec cosec ] + (cosec sec )(cosec + sec ) (sec cosec )( (cosec + sec )) ( + tan cot )( ( + cot + + tan )) (tan cot )(cot + tan ) (tan cot ) cot tan R.H.S. E.: Find the value of cos + cos + cos + cos Sol.: cos + cos + cos + cos cos + cos cos 6 + cos 6 cos + cos + sin + sin cos sin cos sin E.: If tan tan, prove that cos( + ) cos ( ). Sol.: tan tan cos cos sin sin By Componendo and Dividendo Rule, we have cos cos sin sin cos cos sin sin or cos ( ) cos ( ) or cos ( + ) cos ( )

7 E.: Show that sec A seca tan A tan A Sol.: L.H.S. cos A cos A cos A cos A cos A cos A sin sin A cosa A cosa sin A (sin AcosA) sin AcosA sin AcosA sin A sina cosa cos A.tanA sin A cot A. tan A tan A. tan A E.5: If A + C B, prove that : tana tanb tanc tanb tana tanc Sol.: A + C B tan(a + C) tanb or tan A tanc tan A tanc tanb or or tana + tanc tanb tana tanb tanc tana tanb tanc tanb tana tanc E.6: Show that ( + )sin + cos lies between ( 5 ) and ( 5 ). Sol.: We have seen that acos + bsin has limits ± r where r a b. a ; b and r. Since r < 5, the assertion is proved E.7: Show that sin is a root of + 0. Sol.: Let ; 7 ; sin sin cos (sincos) cos cos[cos ] sin( sin ) sin sin sin sin sin sin sin + 0 Hence sin is root of + 0.

8 E.: If < < &, prove that cos cos cot. cos cos Sol.: L.H.S. cos cos cos cos cos cos sin sin cos cos cos sin cos sin sin sin cot R.H. S E.9: Find S n where S n tan tan + tan tan tann tan(n + ) Sol.: Let T r denote the rth term T r tanr tan(r + ) tan[(r + ) r] tan( r ) tan r tan( r ) tan r or tan + tan tan(r + ) tanr tan(r + ) tanr or tan(r + ) tanr cot [tan(r + ) tanr] Putting r,,,..., n and adding, we get S n cot [tan(n +) tan] n cot tan(n +) n cot tan(n +) ( + n) E.0: If, prove that n cos cos cos... cos n : n > n Sol.: Let y n cos cos cos... cos n y sin n sin cos cos cos... cos n n sin cos cos... cos n n sin cos... cos n Repeating this process, y sin sin n sin ( + ) sin y

9 6 E.: Show that cos cos cos Sol.: Let y 6 cos cos cos y sin 7 cos cos cos cos cos cos sin cos cos sin cos 7 7 y sin sin 7 7 sin 7 E.: If sin cos a b a b sin cos, prove that a b ( a b). Sol.: Given sin cos a b a b or b(a + b)sin + a(a + b)cos ab or b(a + b)sin + a(a + b)( sin ) ab or b(a + b)sin + a(a + b)( sin + sin ) ab or (a + b) sin a(a + b)sin + a(a + b) ab 0 or [(a + b)sin a] 0 sin cos a a b b a b Now sin a cos b a a b a b a b b a b ( a b) ( a b)

10 E.: Find the maimum and minimum values of sin 6 + cos 6. Sol.: y sin 6 + cos 6 (sin ) + (cos ) ( sin cos ) ( a + b (a + b) ab(a + b)) sin 5 y ma () 5 y min ( ) E.: If A + B + C, prove that 5 ( cos ) cos cos B + cos C sin A cosa cosb cosc. Sol.: L.H.S. cos B sin A + cos C cos(b + A) cos(b A) + cos C cos( C) cos(a B) + cos C cosc [cos(a B) + cos( A B )] cosc [cos(a B) + cos(a + B)] cosa cosb cosc R.H.S. E.5: If A + B + C, show that Sol.: L.H.S. A B C ( A) ( B ) ( C ) cos cos cos cos cos cos A B C cos cos cos A B A B C cos cos cos C A B C cos cos sin C A B C C cos cos sin cos C A B C cos cos sin C A B C cos cos cos C A B C A B C cos cos cos

11 C B A cos cos cos A B C cos cos cos R.H.S. E.6: If sin sin n c m m 0 c n 0, find the value of n. cos m is an identity in, where c 0, c, c,..., c n are constants and Sol.: sin sin or n c m m 0 cos m sin sin.sin.( sin.sin ). sin n c cos m m m 0 n c cos m m m 0 or or [cos cos ] cos [ cos 6 cos ] cos 6 n c cos m m m 0 n c cos m m m 0 On comparing coefficients of like terms, we get n 6. E.7: Show that cos(sin) > sin(cos) for all belonging to the interval 0,. Sol.: We have to show that sin sin the angles sin and cos, lie in > sin(cos). The sine function increases in 0, for 0,. 0, and Since, sin > cos ( sin + cos < ), cos(sin) > sin(cos)

12 OBJECTIVE QUESTIONS Choose the correct option(s) in the following :. If sin θ and lies in the third quadrant then the value of cos is none of these 5. The value of cos 0º sin 0º is positive negative 0. Which of the following statement is correct? [sin sin C ] sin º > sin sin º < sin sin º sin sin º sin 0. The value of cos º cos º... cos 00º is () 0 none of these 5. sin 0º cos0º is equal to none of these 6. Let A sin 0 + cos, then for all values of, A 0 A 0 < A A 7. sin A sin A is sin A sin A sin A none of these. If sin + sin then cos + cos 0 + cos + cos In a triangle ABC, if angle C is 5º, then ( + cot A) ( + cot B) equals 0. If A + B + C, then cos A + cos B + cos C cosa cosb cosc sina sinb sinc + cosa cosb cosc sina sinb sinc

13 . If + y + z yz, then y y z z yz 0 none of these. If + 60º, then (cos + cos cos cos ) 0 none of these. If cot + tan and sec cos y then sin cos / sin tan y ( y) / (y ) / none of these. The epression cos6 6cos 5cos 0 cos5 5cos 0cos is equal to cos cos cos + cos 5. If sin + cos 7, 0 < <, then tan is equal to 6 7 () ( 7 ) 7 none of these 6. If π 5 sin θ sin θ a for all then the value of a is 5 7 none of these 7. The values of sin lies in the interval 6 [, ] [ /, /] [, 0] none of these. If tan + tan + tan K tan, then K is equal to / none of these 9. If, are two values of obtained from equation a tan + b c sec then the value of α β tan is a/b b/a c/a none of these 0. The minimum value of a sec b tan where a and b are positive, a > b, is a b a b a b none of these

14 . If a cos + b sin c, then (a sin b cos ) c a b c a + b a b + c a + b c. If cosec sin m, sec cos n, then (m n) / + (mn ) / 0. In any triangle ABC the minimum value of tan A + B tan + C tan is. The value of tan 0 + tan 0 + tan 0 tan 0 5. If sin + cos + tan + cot + sec + cosec 7 and sin a b 7, then ordered pair (a, b) can be (6, ) (, ) (, ) (, ) 6. A quadratic equation whose roots are sin, cos 6 are none of these 7. If m tan ( 0) n tan ( + 0), then cos m n ( m n) m n ( m n) m n m n m n m n. In a ABC, cosa sin Bsin C 9. The maimum value of 7 sin 9 cos 7 is 0. The maimum value of + sin cos is 9 MORE THAN ONE CORRECT ANSWERS. If sec tan and y cos ec cot, then y y y y y y + y + 0

15 . The equation sin cos a has a real solution for all value of a a 7 a a 0. If cos ( ) + cos ( ) + cos ( ), then cos 0 sin 0 cossin 0 (cos sin ) 0. If sin cos, then a b a b sin a cos b sin cos a b ( a b) n n 5. If P cos sin, then n sin b sin cos a a ( a b) P6 P P6 P P0 P P P0 5P 0P6 6. If A and B are acute angles such that (A +B) and (A B) satisfy the equation tan tan 0, then A A 6 7. For 0, tan tan tan 0 if B B 6 tan 0 tan 0 tan 0 tan tan. If 0, and cos cos cos( ), then 9. If tan cosec sin,then tan 5 5 (9 5)( 5) (9 5)( 5) 0. The equation sin 6 + cos 6 a has real solutions if a (, ) a, a, none of these

16 Comprehension- MISCELLANEOUS ASSIGNMENT The value of cos cos cos... cos n sin, n n sin n. The value of 6 cos cos cos is If, then the value of cosr is r 6 6. The value of 5 7 sin sin sin is 6 Comprehension- AB is a vertical line and BC is horizontal. D and E are two points on BC. ACB, ADB, AEB. DL and EM are perpendiculars on BC meeting AC at L and M respectively. DL, EM y, BA z.. cot cot cot is equal to z y y z z y none of these 5. cot cot is equal to cot /z y/z y/ /y 6. AD is equal to cot y cot z cot none of these Match the following: 7. A. cos 6 cos 7 (p) 5/ B. cos 6 cos 7 (q) / C. tan 6 tan (r) / D. sin 6 cos (s) (5 5)/5

17 . cos + sin, cos sin y A. cos (p) y B. sin (q) y C. cot (r) tan D. y y (s) y y 9. A. sin cos (p) tan ( / ) tan ( / ) B. sin cos (q) tan C. cos cos (r) tan( / ) tan( / ) D. cos (s) tan( / ) tan( / ) INTEGER TYPE QUESTIONS 0. tan 6 9 tan tan 9 is equal to. If,, then cos sin sin is always equal to. If sin y, then must be k. If tan tan +, then cos + sin. cos 6 cos7 cos0 cos, 6 then is equal to 5. If 9 5y 56 cos sin 9sin 5y cos and 0, then value of cos sin [(9 ) (5 y) ] 7 / / 6. If, are positive acute angles and cos cos cos, then tan k tan, then k 7. If. If sin sin sin 6, then cos cos cos tan tan tan, then tan 9. If sin 7 sin 6 sin sin 5 cos, then

18 PREVIOUS YEAR QUESTIONS IIT-JEE/JEE-ADVANCE QUESTIONS. Which of the following number(s) is/are rational? sin 5 cos 5 sin 5 cos 5 sin 5 cos 75. For 0 < < if n n n n then cos, y sin, z cos sin n 0 n 0 n 0 yz z + y yz y + z yz + y + z yz yz +. For a positive integer n, let f n () tan ( + sec ) ( + sec ) ( + sec )... ( + sec n ), then f 6 f f 6 All of these. If + and +, then tan equals (tan + tan ) tan + tan tan + tan tan + tan 5. The maimum value of cos cos... cos n under the restrictions 0,,,..., n and cot cot cot... cot n is n / n n P Q 6. In a triangle PQR, R. If tan and tan are the roots of the equation a + b + c 0 (a 0) then a + b c b + c a a + c b b c 7. Let n be an odd integer. If sin n n r 0 b r sin r, for every value of, then b 0, b b 0 0, b n b 0, b n b 0 0, b n n +. sec y ( y) is true if and only if + y 0 y, 0 y 0, y 0

19 The value of cos cos cos cos is equal to cos 0. The value of the epression cosec 0 sec 0 is equal to sin 0 sin 0 sin 0 sin 0. The graph of the function cos cos( + ) cos ( + ) is a straight line passing through (0, sin ) with slope a straight line passing through (0, 0) a parabola with verte (, sin ) a straight line passing through the point, sin and parallel to the -ais. For 0 < < if n 0 cos n, y sin n 0 n ; z cos n 0 n sin n, then yz z + y yz y + z yz + y + z yz yz +. If in the triangle PQR, sin P, sin Q, sin R are in A.P., then the altitudes are in A.P. the altitudes are in H.P. the medians are in G.P the medians are in A.P.. If sin cos, then 5 tan sin cos 7 5 tan sin cos For 0, the solution(s) of 6 ( m ) m cosec cosec is(are) m 6 5

20 6. The number of all possible values of, where 0 < <, for which the system of equations cos sin (y + z) cos (yz) sin sin y z (yz) sin (y + z) cos + y sin have a solution ( 0, y 0, z 0 ) with y 0 z 0 0, is 7. Let P { :sin cos cos } and Q { :sin cos sin } be two sets. Then P Q and Q P Q Q P P Q P Q. Let, [0, ] be such that cos ( sin ) sin tan cot cos, tan( ) 0 and sin. Then cannot satisfy 0 9. Match List I with List II and select the correct answer using the code given below the lists : List I P. cos(tan y) ysin(tan y) y cot(sin y) tan(sin y) y / takes value. 5 Q. If cos + cos y + cos z 0 sin + sin y + sin z then. y possible value of cos is R. If cos cos sin sin sec cos sin sec. cos cos then possible value of sec is S. If cot sin sin tan 6, 0,. then possible value of is P-(); Q-(); R-(); S-() P-(); Q-(); R-(); S-() P-(); Q-(); R-(); S-() P-(); Q-(); R-(); S-()

21 . If sin A sin B and cos A cos B, then A DCE QUESTIONS n + B n B n + B n + ( ) n B. tan 0 + tan 5 + tan 0 tan 5 0. If cos ( + ) m cos ( ), tan is [( + m)/( m)]tan [( m)/( + m)]tan [( m)/( + m)]cot [( + m)/( m)]sec. If cos + cos cos y, sin + sin sin y, then the value of cos is If tan tan tan K tan, then the value of K is none of these 6. If cos 0 K and cos K, then the possible values of between 0 and 60 are 0 0 and 0 0 and 0 50 and 0 7. The maimum value of sin 6 sin is 5 none of these. The value of cos 7 cos 7 cos 6 7 cos 7 7 is 9. If cos + cos a, sin + sin b, then cos( ) is equal to ab a b a b a b a b b a a b 0. If cos A, then value of sin A 5A sin is None of these

22 5 7. cos cos cos cos is equal to cos ( ) AIEEE/JEE-MAINS QUESTIONS. If 0 < <, and cos + sin, then tan is 7 / 7 / 7 / 7 /. Let, be such that < <. If sin + sin and cos 65 + cos 7, then the 65 value of cos 0 is The sides of a triangle are sin, cos and sin cos for some 0 < <. Then the greatest angle of the triangle is The number of values of in the interval [0, ] satisfying the equation sin + 5 sin 0 is 6 5. Let A and B denote the statements A : cos + cos + cos 0 B : sin + sin + sin 0 If cos ( ) + cos ( ) + cos ( ), then : A is false and B is true both A and B are true both A and B are false A is true and B is false 5 6. Let cos ( ) and let sin ( ), where 0 <,. Then tan If A sin + cos, then for all real : A 6 A A A 6

23 . In a PQR, if sin P + cos Q 6 and sin Q + cos P, then the angle R is equal to If, y, z are in A.P. and tan, tan y and tan z are also in A.P., then 6 y z 6 y z y z y 6z tan A cot A 0. The epression can be written as cot A tan A tana + cota seca + coseca sina cosa + seca coseca +. ABCD is a trapezium such that AB and CD are parallel and BC CD. If ADB, BC p and CD q, then AB is equal to p q p cos q sin p q sin p cos q sin p q sin p cos q sin p q cos p cos q sin. Let f K () K (sink + cos k ) where R and K. Then f () f 6 () equals: 6

24 . Prove the following identities sin (n + )A sin na sin(n + )A sin A cos cos + sin ( ) sin ( + ) cos ( + ) sin + cos ( + )sin sin + cos ( + ) cos tan A tan A tan A tan A tan A tan A (e) sin 0 sin 0 sin 50 sin 70. Prove that BASIC LEVEL ASSIGNMENT 6. cosec cosec sec sec sec tan sin sin tan tan tan tan tan tan. Prove that 5 sin sin sin tan 0 tan 0 tan 60 tan 0 cos cos cos cos cos cos Prove that sin sin + cos cos cos. 5. Prove that cos A + cos A cos A 6. Show that : sin 6º + sin º + sin º sin º + sin 90º. 7. Prove that : tan 70º tan 50º + tan 0º.. If A + B 5º, show that : ( + tana) ( + tanb).

25 9. Prove that sin A sina sin5a sin7 A cos A cosa cos5a cos7a tan A. 0. Prove that : tan + tan + tan + cot cot.. If A + B + C, then prove that : cos A sin B sinc cos B cos C. sin C sin A sin A sin B. If sin sin cos cos + 0, prove that + cot tan 0.. Find the value of cos º.. The arc of a circle of radius cm subtends an angle of 60º at the centre. Find the length of the arc. (Take /7). 5. The angles of a triangle are in A.P. and least angle is 0º ; epress the greatest angle in radians. 6. If tana + sina m and tana sina n, then shown that m n mn. 7. If cos a, then prove that cos a a. a. If sin + sin y a and cos + cos y b, then prove that cos( y) a b and tan y a b a b. y 9. If tan tan then prove that sin y sin sin sin 0. If cot cot ( ) cot ( ), show that cot (cot + cot )

26 ADVANCED LEVEL ASSIGNMENT. If tan θ e e tan, show that cosθ e cos ecosθ. Sum the series cosec + cosec + cosec +... to n terms.. Prove that sin + sin + sin sin(n ) sin n. sin. Show that tan + tan + tan + tan +... to n + terms cot n cot n 5. If, show that tan tan + tan tan + tan tana 7. 7 cos sin cos y sin y 6. If, then prove that. cos y sin y cos sin 7. If A + B + C, prove that (tana + tanb + tanc )(cota + cotb + cotc ) + seca secb secc.. Determine the smallest positive value of (in degrees) for which tan( + 00º) tan( + 50º) tan tan( 50º). y y z z 9. If + y + z y z, show that y z y y z z.. y z. 0. Show that sin cos sin cos 9 sin 9 cos 7 [tan 7 tan ].. If A + B + C epress S sina + sinb + sinc as a product of three trigonometric ratios. If S 0, show that at least one of the angles is 60º.. Find a and b such that a cos + 5 sin b for all.. Show that the value of tan tan wherever defined never lies between and.

27 . If 0 < <, prove that cot + cot. 5. If tan ntan (n > 0), prove that tan ( ) ( n ) n. 6. If cos m and tan tan, prove that cos sin m. 7. If 0 < <, 0 < < and cos cos cos( + ), prove that.. If 0 < <, prove that tan sin tan sin tan cos. 9. If A + B + C, prove that cos cos B C B C C A cos C A cos cos cos A B A B If A, B and C are in Arithmetic progression, determine the values of A, B and C where ABC is a triangle such that sin(a + B) sin(c A) sin(b + C)

28 ANSWERS Objective Questions (b,c). (b,c,d). (a,b,d). (a,c,d) 5. (a,d) 6. (a,d) 7. (c,d). (a,b,c) 9. (b,c) 0. (b,c) Miscellaneous Assignment A-(q); B-(r); C-(s); D-(p). A-(q); B-(p); C-(s); D-(r) 9. A-(r); B-(s); C-(q); D-(p) 0. (). (). (). (0). () 5. () 6. () 7. (). () 9. (7) Previous Year Questions ANSWERS FOR IIT-SCREENING (a,b) 5. (c,d) (a,c,d) 9. ANSWERS FOR DCE

29 ANSWERS MAINS QUESTIONS Basic Level Assignment.. cm 5. / radians Advanced Level Assignment. cot θ cot n. 0º. A cos cos 0. 5º, 60º, 75º B C cos. a 5 and b 5

[STRAIGHT OBJECTIVE TYPE] Q.4 The expression cot 9 + cot 27 + cot 63 + cot 81 is equal to (A) 16 (B) 64 (C) 80 (D) none of these

[STRAIGHT OBJECTIVE TYPE] Q.4 The expression cot 9 + cot 27 + cot 63 + cot 81 is equal to (A) 16 (B) 64 (C) 80 (D) none of these Q. Given a + a + cosec [STRAIGHT OBJECTIVE TYPE] F HG ( a x) I K J = 0 then, which of the following holds good? (A) a = ; x I a = ; x I a R ; x a, x are finite but not possible to find Q. The minimum value