Problem 24 - Precalculus Essentials

Question

Verify each identity. $$\frac{1-\sin \theta}{\cos \theta}=\sec \theta-\tan \theta$$

Step-by-Step Solution

Answer

The given trigonometric identity is verified to be correct. \(\frac{1-\sin \theta}{\cos \theta} = \sec \theta - \tan \theta\).

Step 1: Write down the problem

The problem to solve is: \(\frac{1-\sin \theta}{\cos \theta} = \sec \theta - \tan \theta\)

Step 2: Start with Left Side

First, start with the left side of the equation. Rewrite it to the form of trigonometric identities. Thus, \(\frac{1-\sin \theta}{\cos \theta} =\frac{1}{\cos \theta}-\frac{\sin \theta}{\cos \theta}\)

Step 3: Apply Trigonometric Identities

Now replace \(\frac{1}{\cos \theta}\) with \(\sec \theta\) and \(\frac{\sin \theta}{\cos \theta}\) with \(\tan \theta\). Now left side becomes \(\sec \theta - \tan \theta\)

Step 4: Compare Both Sides

Since both sides of the identity are the same, that is, \(\sec \theta - \tan \theta = \sec \theta - \tan \theta\), it proves that the given identity is correct.