Answer
$\cos{\theta}\cot{\theta}=\frac{1}{\sin\theta}-\sin\theta$
Refer to the step-by-step part for the proof.
Work Step by Step
Step 1: Replace $\cot\theta$ by $\frac{\cos\theta}{\sin\theta}$
Hence,
$\cos\theta\cot\theta=\cos\theta\times \frac{\cos\theta}{\sin\theta}=\frac{\cos^{2}\theta}{\sin\theta}$
Step 2: By the identity $\sin^{2}\theta+\cos^{2}\theta=1$; replace $\cos^{2}\theta$ by $1-\sin^{2}\theta$.
Hence,
$\cos\theta\cot\theta=\frac{1-\sin^{2}\theta}{\sin\theta}$
Step 3: Split the denominator across common numerator,
$\cos\theta\cot\theta=\frac{1}{\sin\theta}-\frac{\sin^{2}\theta}{\sin\theta}$
Step 4: Cancel one $\sin\theta$ in the second term,
Hence,
$\cos\theta\cot\theta=\frac{1}{\sin\theta}-\sin\theta$
