Question

Prove the following trigonometric identities.

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

Answer 1

In the given question, we need to prove .

Using the property , we get

So,

$$\begin{gathered} \frac{1+\cos \theta -{\sin }^2\theta }{\sin \theta \left(1+\cos \theta \right)} \\ =\frac{1+\cos \theta -\left(1-{\cos }^2\theta \right)}{\sin \theta \left(1+\cos \theta \right)} \\ =\frac{\cos \theta +{\cos }^2\theta }{\sin \theta \left(1+\cos \theta \right)} \end{gathered}$$

Solving further, we get

Hence proved.