BRAIN TEASURES TRIGONOMETRICAL RATIOS BY ABHIJIT KUMAR JHA EXERCISE I. or tan &, lie between 0 &, then find the value of tan 2.

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1 EXERCISE I Q Prove that cos² + cos² (+ ) cos cos cos (+ ) ² Q Prove that cos ² + cos (+ ) + cos (+ ) Q Prove that, ta + ta + ta + cot cot Q Prove that : (a) ta 0 ta 0 ta 60 ta 0 (b) ta 9 ta 7 ta 6 + ta (c) Q Calculate without ug trigoometric tables : 6 (a) cosec 0 sec 0 (b) cos 0 cot 0 (c) cos 0 cos0 0 sec cos0 7 (d) 0 (e) cos 6 + cos 6 + cos 6 + cos (f) ta 0 ta 0 + ta 70 7 Q6(a) If X 7 + +, Y cos + cos + cos X Y the prove that ta Y X (b) Prove that ² + ² + ² 9 + ² + ² 9 + ² Q7 Show that : (a) cot 7 or ta or 6 (b) ta + 6 Q If m ta (- 0 ) ta (+ 0 ), show that cos m (m) y Q9 If ta Q0 If cos (+ ) ta, prove that y ; (- ) &, lie betwee 0 &, the fid the value of ta ta ta Q Prove that if the agles & satisfy the relatio m the m Q (a) If y 0 cos² 6 cos + ², the fid the greatest & least value of y (b) If y + + cos, fid the maimum & miimum values of y R (c) If y 9 sec + 6 cosec, fid the miimum value of y R (d) Prove that cos (e) Prove that + cos + lies from - & 0 + cos lies betwee & tata m m taa Q If A + B + C, prove that tabtac (ta A) (cot A)

2 Q If + c where, > 0 each lyig betwee 0 ad / ad c is a costat, fid the maimum or miimum value of (a) + (b) (c) ta + ta (d) cosec + cosec Q Let A, A,, A be the vertices of a -sided regular polygo such that ; A Fid the value of A A A A A Q6 Prove that : cosec + cosec + cosec + + cosec cot (/) cot - Q7 For all values of,, prove that; cos + cos + cos+ cos (+ + ) cos cos A cosb A B Q Show that cosa B (A B) cosa cosb Q9 If ta ta ta ta ta, prove that cos Q0 If +, prove that cos²+ cos² + cos² + cos cos cos Q If + + ta ta ta, show that ta ta ta Q If A + B + C ad cot cot A + cot B + cot C, show that, (A ) (B ) (C ) 7 Q If P cos cos cos cos ad Q cos cos cos cos, the fid P Q cos cos cos Q If A, B, C deote the agles of a triagle ABC the prove that the triagle is right agled if ad oly if A + B + C 0 Q Give that ( + ta )( + ta )( + ta ), fid EXERCISE II Q If ta p/q where 6, beig a acute agle, prove that; (p cosec q sec ) p q Q Let A, A, A A are the vertices of a regular sided polygo iscribed i a circle of radius R If (A A ) + (A A ) + + (A A ) R, fid the umber of sides i the polygo Q Prove that: cos cos (cos + cos) cos() ( + ) () cos( ) Q Without ug the surd value for 0 or cos 6 0, prove that 6 0 cos 0

3 Q Show that, 9 (ta7 ta) cos cos9 cos7 r r Q6 Let cos ad r cos, the show that r cos ec, where deotes the cotiued product 6 Q7 If, prove that ta ta + ta ta + ta ta 7 7 Q For 0 < < cos prove that, > (cos) Q9 (a) If prove that, Q 0 Let k, the prove that 0 cos k cos( cosk ) k k (b) Q Prove that the value of cos A + cos B + cos C lies betwee & where A + B + C Q If cosa tab, cosb tac ad cosc taa, the prove that A B C cos Q Show that R ca ot have ay value betwee ad What iferece ca you draw about the values of? cos Q If ( + t)( + cos t) Fid the value of ( t)( cos t) cos Q Prove that from the equality follows the relatio ; a b ab a cos b Q6 Prove that the triagle ABC is equilateral iff, cot A + cot B + cot C Q7 Prove that the average of the umbers,,, 6,, 0, is cot Q Prove that : 7 / / ab A Q9 If A+B+C ; prove that ta B C + ta + ta Q0 If A+B+C (A, B, C > 0), prove that A B C Q Show that ellimiatig & y from the equatios, + y a ; cos + cos y b & ta + ta y c gives a b ab a Q Determie the smallest positive value of (i degrees) for which ta( + 00 ) ta( + 0 ) ta ta ( 0 ) ta Q Evaluate : cos c

4 Q If + + & ta ta ta, the prove that; + cos + cos + cos 0 Q R, fid the rage of the fuctio, f () cos ( + ) ; [0, ] EXERCISE III Q sec y (y) is true if ad oly if : (A) + y 0 (B) y, 0 (C) y (D) 0, y 0 [JEE 96, ] Q (a) Let be a odd iteger If r 0 b r r, for every value of, the : (A) b 0, b (B) b 0 0, b (C) b 0, b (D) b 0 0, b + (b) Let A 0 A A A A A be a regular heago iscribed i a circle of uit radius The the product of the legths of the lie segmets A 0 A, A 0 A & A 0 A is : (A) (B) (C) (D) (c) Which of the followig umber(s) is/are ratioal? (A) º (B) cos º (C) º cos º (D) º cos 7º [ JEE '9, out of 00 ] Q For a positive iteger, let f () ta (+ sec ) (+ sec ) (+ sec ) ( + sec ) The (A) f (B) f 6 (C) f (D) f 6 Q(a) Let f () ( + ) The f () : (A) 0 oly whe 0 (B) 0 for all real (C) 0 for all real (D) 0 oly whe 0 [ JEE 000 Screeig out of ] (b) I ay triagle ABC, prove that, cot A + cot B + cot C cot A cot B cot C Q(a) Fid the maimum ad miimum values of 7 cos [ JEE 000 Mais, out of 00 ] [JEE '99,] (b) Fid the smallest positive values of & y satisfyig, y, cot + cot y [REE 000, ] Q6 If + ad + the ta equals (A) (ta + ta) (B) ta + ta (C) ta + ta (D) ta + ta [ JEE 00 (Screeig), out of ] Q7 If ad are acute agles satisfyig, cos, the + (A), (B), (C), 6 (D), 6 [JEE 00 (Screeig)]

5 Q I a equilateral triagle, cois of radii uit each are kept so that they touch each other ad also the sides of the triagle Area of the triagle is (A) + (B) 6 + (C) + 7 (D) + 7 [JEE 00 (Screeig)] Q9 Let 0, ad t (ta) ta, t (ta) cot, t (cot) ta, t (cot) cot, the (A) t > t > t > t (B) t > t > t > t (C) t > t > t > t (D) t > t > t > t [JEE 006, ] ANSWER SHEET EXERCISE I Q (a) (b) (c) (d) (e) (f) Q 0 6 Q (a) y ma ; y mi (b) y ma ; y mi, (c) 9 Q (a) ma (c/), (b) ma (c/), (c) mi ta (c/), (d) mi cosec (c/) Q 7 Q Q Q 7 Q, Q EXERCISE II Q 0 Q 0 Q y EXERCISE III Q B Q (a) B, (b) C, (c) C Q A, B, C, D Q (a) C Q (a) ma & mi ; (b) ; y Q6 C Q7 B 6 Q B Q9 B

Objective Mathematics

Objective Mathematics 6. If si () + cos () =, the is equal to :. If <