Prove each of the following iden tities . A. (1-sin theater)(1+sin theater)/sin square theater= cot square theater. B.sin theater/ 1- cos square theater= cosec theater. C.( sec theater- tan theater)( sec theater + tan theater=1 D. (cosec theater - cot theater )( cosec theater + cot theater = 1. E.cos raised to the power of 4 theater - sin raised to the power of 4 theater = 2cos square theater - 1

Answer 1

A.

(1-sinθ)(1+sinθ)/sin²θ = (1-sin²θ)/sin²θ = cos²θ/sin²θ = cot²θ

B.

sinθ/1-cos²θ = sinθ/sin²θ = 1/sinθ = cosecθ

C.

(secθ-tanθ)(secθ+tanθ) = sec²θ - tan²θ = 1/cos²θ - sin²θ/cos²θ = (1 - sin²θ)/cos²θ = cos²θ/cos²θ = 1

D.

(cosecθ-cotθ)(cosecθ+cotθ) = cosec²θ - cot²θ = 1/sin²θ - cos²θ/sin²θ = (1 - cos²θ)/sin²θ = sin²θ/sin²θ = 1

E.

cos⁴θ - sin⁴θ = (cos²θ + sin²θ)(cos²θ - sin²θ) = cos²θ(1 - sin²θ) - sin²θ(1 - cos²θ) = cos²θ - cos⁴θ - sin²θ + sin⁴θ = 2cos²θ - 1 - 2sin²θ = 2(cos²θ - sin²θ) - 1 = 2cos²(θ) - 1