Question

Prove that `(cos θ - 2 cos^3 θ)/(sin θ - 2 sin^3 θ) + cot θ = 0`.

Answer 1

L.H.S. = `(cos θ - 2 cos^3 θ)/(sin θ - 2 sin^3 θ) + cot θ`

= `(cos θ(1 - 2 cos^2 θ))/(sin θ (1 - 2 sin^2 θ)) + cot θ`

= `cot θ ((sin^2 θ + cos^2 θ - 2 cos^2 θ))/((sin^2 θ + cos^2 θ - 2 sin^2 θ)) + cot θ`

= `- cot θ ((sin^2 θ - cos^2 θ))/((cos^2 θ - sin^2 θ)) + cot θ`

= −cot θ + cot θ

= 0

L.H.S. = R.H.S.

Hence proved.