Question

Prove the following trigonometric identities:
1+cosθ-sin²θ/sinθ(1+cosθ)=cotθ

Answer 1

The identity $\dfrac{1+\cos \theta-\sin^2 \theta}{\sin \theta(1+\cos \theta)}=\cot \theta$ , proved.

Step-by-step explanation:

To prove that, the identity: $\dfrac{1+\cos \theta-\sin^2 \theta}{\sin \theta(1+\cos \theta)}=\cot \theta.$

L.H.S. = $\dfrac{1+\cos \theta-\sin^2 \theta}{\sin \theta(1+\cos \theta)}$

= $\dfrac{(1-\sin^2 \theta)+\cos \theta}{\sin \theta(1+\cos \theta)}$

Using the trigonometric identity,

$\sin^2 A+\cos^2 A$ = 1

⇒ $\cos^2 A=1-\sin^2 A$

= $\dfrac{\cos^2 \theta+\cos \theta}{\sin \theta(1+\cos \theta)}$

Taking $\cos \theta$ as common in numerator part, we get

= $\dfrac{\cos \theta(1+\cos \theta)}{\sin \theta(1+\cos \theta)}$

= $\dfrac{\cos \theta}{\sin \theta}$

Using the trigonometric identity,

$\cot A=\dfrac{\cos A}{\sin A}$

= $\cot \theta$

= R.H.S., proved,

Thus, the identity $\dfrac{1+\cos \theta-\sin^2 \theta}{\sin \theta(1+\cos \theta)}=\cot \theta$ , proved.