## Precalculus (6th Edition) Blitzer

The left side of the expression $\frac{\sin x}{\tan x}+\frac{\cos x}{\cot x}$ can be further simplified by using the quotient identities $\tan x=\frac{\sin x}{\cos x}$ and $\cot x=\frac{\cos x}{\sin x}$. \begin{align} & \frac{\sin x}{\tan x}+\frac{\cos x}{\cot x}=\frac{\sin x}{\frac{\sin x}{\cos x}}+\frac{\cos x}{\frac{\cos x}{\sin x}} \\ & =\sin x.\frac{\cos x}{\sin x}+\cos x.\frac{\sin x}{\cos x} \\ & =\cos x+\sin x \end{align} Hence, the left side is equal to the right side $\frac{\sin x}{\tan x}+\frac{\cos x}{\cot x}=\sin x+\cos x$.