Chapter C, Problem 23E

To determine

To prove:

  $\sin \theta \cot \theta =\cos \theta$

Explanation of Solution

Given information:

Eq: $\sin \theta \cot \theta =\cos \theta$

Formula Used:

  $\cot \theta =\frac{1}{\tan \theta }$

Proof:

The equation is given as:

  $\sin \theta \cot \theta =\cos \theta$

The left hand side of the eq. which should be give result as right hand side:

  $\begin{array}{l}\sin \theta \cot \theta =\sin \theta \times \frac{1}{\tan \theta }\\ \sin \theta \cot \theta =\sin \theta \times \frac{1}{\left(\frac{\sin \theta }{\cos \theta }\right)}\left[\because \tan \theta =\frac{\sin \theta }{\cos \theta }\right]\\ \sin \theta \cot \theta =\sin \theta \times \frac{\cos \theta }{\sin \theta }\\ \sin \theta \cot \theta =\cos \theta \end{array}$

So, $\sin \theta \cot \theta =\cos \theta$ is an identity for all values of $\theta$

Hence, proved.