Chapter 6 - Section 6.3 - Trigonometric Equations Involving Multiple Angles - 6.3 Problem Set - Page 342: 63

$$\frac{1}{1+\cos t}+\frac{1}{1-\cos t} =2\csc^2 t $$

Since \begin{align*} \frac{1}{1+\cos t}+\frac{1}{1-\cos t}&=\frac{1-\cos t+1+\cos t}{(1+\cos t)(1-\cos t)}\\ &=\frac{2 }{ 1-\cos^2 t },\ \ \text{use}\ \ \cos ^2x+\sin ^2x=1 \\ &=\frac{2 }{ \sin^2t }\\ &=2\csc^2 t \end{align*} Then $$\frac{1}{1+\cos t}+\frac{1}{1-\cos t} =2\csc^2 t $$