Question

If cos² θ - sin² θ = tan² φ, then prove that cos φ = 1/√2 cos θ​

Answer 1

Step-by-step explanation:

Given (cos2θ−sin2θ)=tan2ϕ

To prove:

cosϕ=2cosθ1

(cos2θ−sin2θ)=tan2ϕ

add 1 on both sides

(cos2θ−sin2θ)=tan2ϕ+1

cos2+(1−sinθ)=(1+tan2ϕ)

cos1θ+cos2θ=sec2ϕ

⇒ 2cos2θ=sec2ϕ

⇒ 2cos2θcos2ϕ=1

⇒ cos2ϕ=2cos1θ1

⇒ cosθ=1/root2 cos o