## Precalculus (6th Edition) Blitzer

The given expression on the left side $\frac{\cos t}{\cot t-5\cos t}$ can be simplified by multiplying and dividing the terms by $\frac{1}{\cos t}$ and apply the quotient identity $\cot t=\frac{\cos t}{\sin t}$ for further simplification. \begin{align} & \frac{\cos t}{\cot t-5\cos t}=\frac{\cos t}{\cot t-5\cos t}.\frac{\frac{1}{\cos t}}{\frac{1}{\cos t}} \\ & =\frac{\frac{\cos t}{\cos t}}{\frac{\cot t-5\cos t}{\cos t}} \\ & =\frac{1}{\frac{\cot t}{\cos t}-\frac{5\cos t}{\cos t}} \\ & =\frac{1}{\frac{\cos t}{\sin t}.\frac{1}{\cos t}-5} \end{align} And now the expression can be further simplified by using the reciprocal identity $\frac{1}{\sin t}=\csc t$ \begin{align} & \frac{1}{\frac{\cos t}{\sin t}.\frac{1}{\cos t}-5}=\frac{1}{\frac{1}{\sin t}-5} \\ & =\frac{1}{\csc t-5} \end{align} Hence, the expression on the left side is equal to the expression on the right side.