Question

Establish each identity.

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

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

Step 2. Calculation

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

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

Hence, Proved