Please prove the following identities:?

2013-03-09 10:03 am

a. tan2θ = csc4θ - cot4θ
b. (cos^2θ+ secθ)/ (1+ secθ) = 2- sin^2θ - cosθ
c. (1-cosθ) csc2θ cot(θ/2)= (1/2)secθ

I already tried (a) answer always turns out to be 3/2 tan2θ
and the other two are very complicated, please help me understand these...

回答 (1)

An ESL Learner

2013-03-09 10:25 am

✔ 最佳答案

a)
csc4θ - cot4θ
= 1/sin4θ - cos4θ/sin4θ
= (1 - (1 - 2sin^2θ))/sin4θ
= 2sin^2θ/(2sinθcos2θ)
= sin2θ/cos2θ
= tan2θ

b)
(cos^2θ + secθ)/(1 + secθ)
= (cos^2θ + 1/cosθ)/(1 + 1/cosθ)
= (cos^3θ + 1^3)/(cosθ + 1)
= (cosθ + 1)(cos^2θ - cosθ + 1)/(cosθ + 1)
= (1 - sin^2θ) - cosθ + 1
= 2 - sin^2θ - cosθ

c)
(1 - cosθ) * csc2θ * cot(θ/2)
= (1 - cosθ) * 1/sin2θ * 1/(sin(θ/2)/cos(θ/2))
= (1 - cosθ) * 1/(2sinθcosθ) * 1/(2sin(θ/2)cos(θ/2) / 2cos^(θ/2))
= (1 - cosθ) * 1/(2sinθcosθ) * 1/(sinθ / (2cos^(θ/2) - 1 + 1))
= (1 - cosθ) * 1/(2sinθcosθ) * (cosθ + 1)/sinθ
= (sin^2θ) * 1/(2sin^2θcosθ)
= 1/(2cosθ)
= 1/2 * secθ

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