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Trigonometry - Trigonometric identities.
Test Yourself 1.


Simplify the following trigonometric identities:

Basic substitutions. 1. Simplify sinβ cosecβ
Answer.1.		 2. Simplify tanβ secβ cosecβ
Answer.sec2x.
  3. 2sinθ secθ - 3 tanθ
Answer.-tanθ.		 4. cot A sec A sin A
Answer.1.
  5.
Answer.cosec2 A.		 6.
Answer.2sin α
  7.

Answer.cot x

 8. cos2A × tan2A + cos2A
Answer.1
Complementary angles. 9.
Answer.sinθ cos θ		 10.
Answer.2cot2 x
Pythagorean relationships. 11. If x = sin α express

1 - cot2 α in terms of x.

Answer.2-1/(x2).		 12. 4 cosec2 θ - 4 cot2 θ
Answer.4
  13.

Answer.sin β

 14. tan α cosec α.
Answer.sec α

Prove the following trigonometric identities:
Basic substitutions
(change of defintion).		 15. 16.
  17. (sec2 α - 1) cos2 α = sin2 α 18. sec2 x - tan2 x = 1
Complementary angles. 19.

20. Simplify .
Pythagorean relationships. 21. (1 - sin θ)(1 + sin θ) = cos 2θ 22. tan2θ cos2θ + cot2θ sin2θ = 1
  23. sin4 x + cos2x sin2x = sin2 x 24. sec β - cos β = sin β tan β
  25. 26.
  27. 3sin2 A + 2 cos2 A = sin2 A + 2 28.
  29. 30.
  31.
  32. 33.
  34. 35. Prove that

(sin B + cos B)2 + (sin B - cos B)2 = 2