Question

In problems 19-98, establish each identity.

$\frac{\sec \left(\theta \right)-\csc \left(\theta \right)}{\sec \left(\theta \right)\csc \left(\theta \right)}=\sin \left(\theta \right)-\cos \left(\theta \right)$

Short answer

We have proved the identity$\frac{\sec \left(\theta \right)-\csc \left(\theta \right)}{\sec \left(\theta \right)\csc \left(\theta \right)}=\sin \left(\theta \right)-\cos \left(\theta \right)$.

Step-by-step solution

Step 1. Given information.

We have:

$\frac{\sec \left(\theta \right)-\csc \left(\theta \right)}{\sec \left(\theta \right)\csc \left(\theta \right)}=\sin \left(\theta \right)-\cos \left(\theta \right)$

Step 2. Establish the identity.

Take the left hand side:

$$\begin{gathered} \frac{\sec \left(\theta \right)-\csc \left(\theta \right)}{\sec \left(\theta \right)\csc \left(\theta \right)} \\ =\frac{{\displaystyle \frac{1}{\cos \left(\theta \right)}}-{\displaystyle \frac{1}{\sin \left(\theta \right)}}}{\frac{1}{\cos \left(\theta \right)}\times \frac{1}{\sin \left(\theta \right)}} \\ =\frac{{\displaystyle \frac{\sin \left(\theta \right)-\cos \left(\theta \right)}{\cos \left(\theta \right)\sin \left(\theta \right)}}}{{\displaystyle \frac{1}{\cos \left(\theta \right)\sin \left(\theta \right)}}} \\ =\frac{\sin \left(\theta \right)-\cos \left(\theta \right)}{\cos \left(\theta \right)\sin \left(\theta \right)}\times \frac{\cos \left(\theta \right)\sin \left(\theta \right)}{1} \\ =\sin \left(\theta \right)-\cos \left(\theta \right) \end{gathered}$$

Hence, proved.