Question

Prove the following trigonometric identities:

$\frac{1+cos\theta -si{n}^2\theta }{sin\theta (1+cos\theta )}=cot\theta$

Answer 1

Ans:

To prove,

1+cosΘ−sin2ΘsinΘ(1+cosΘ)1+cosΘ−sin2ΘsinΘ(1+cosΘ) = cot ΘΘ

Considering left hand side (LHS),

= 1+cosΘ−(1−cos2Θ)sinΘ(1+cosΘ)1+cosΘ−(1−cos2Θ)sinΘ(1+cosΘ)

= 1+cosΘ−1+cos2ΘsinΘ(1+cosΘ)1+cosΘ−1+cos2ΘsinΘ(1+cosΘ)

= cosΘ+cos2ΘsinΘ(1+cosΘ)cosΘ+cos2ΘsinΘ(1+cosΘ)

= cosΘ(1+cosΘ)sinΘ(1+cosΘ)cosΘ(1+cosΘ)sinΘ(1+cosΘ)

= (cosΘ)(sinΘ)(cosΘ)(sinΘ)

= cotΘcotΘ

Therefore, LHS = RHS

Hence, proved.