Answer
$\csc x-\sin x=\cos x\cot x$
Work Step by Step
$\csc x-\sin x=\cos x\cot x$
On the left side of the equation, substitute $\csc x$ with $\dfrac{1}{\sin x}$:
$\dfrac{1}{\sin x}-\sin x=\cos x\cot x$
Evaluate the difference of fractions on the left side:
$\dfrac{1-\sin^{2}x}{\sin x}=\cos x\cot x$
Substitute $1-\sin^{2}x$ with $\cos^{2}x$:
$\dfrac{\cos^{2}x}{\sin x}=\cos x\cot x$
Rewrite the expression on the left like this:
$\cos x\Big(\dfrac{\cos x}{\sin x}\Big)=\cos x\cot x$
Since, $\dfrac{\cos x}{\sin x}=\cot x$, the identity is proved:
$\cos x\cot x=\cos x\cot x$
