Question

Establish each identity.

$\frac{\sin \theta +\cos \theta }{\cos \theta }-\frac{\sin \theta -\cos \theta }{\sin \theta }=sec\theta csc\theta$

Short answer

The left side of the given equation has been simplified to bring the right side.

Step-by-step solution

Step 1. Given Information

The given equation is$\frac{\sin \theta +\cos \theta }{\cos \theta }-\frac{\sin \theta -\cos \theta }{\sin \theta }=sec\theta csc\theta$

Step 2. Calculation

Consider the left side of the given equation and simplify to bring the right side in order to establish the equation.

$$\begin{gathered} \frac{\sin \theta +\cos \theta }{\cos \theta }-\frac{\sin \theta -\cos \theta }{\sin \theta }=\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\cos \theta }-\frac{\sin \theta }{\sin \theta }+\frac{\cos \theta }{\sin \theta } \\ =\frac{\sin \theta }{\cos \theta }+1-1+\frac{\cos \theta }{\sin \theta } \\ =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta } \\ =\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta } \\ =\frac{1}{\sin \theta \cos \theta } \\ =sec\theta csc\theta \end{gathered}$$

Hence, Proved