Question

Establish each identity.

$sec\theta -\cos \theta =\sin \theta \tan \theta$

Short answer

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

Step-by-step solution

Step 1. Given Information

The given equation is$sec\theta -\cos \theta =\sin \theta \tan \theta$

Step 2. Calculation

Consider the left side of the given equation and simplify.

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

Thus, the given identity is established.