Problem 14 - Precalculus Essentials

Question

Verify each identity. $$\frac{\cos \theta \sec \theta}{\cot \theta}=\tan \theta$$

Step-by-Step Solution

Answer

The left side of the equation simplifies to \( \tan \theta \), which matches the right side of the equation, hence verifying the identity \( \frac{\cos \theta \sec \theta}{\cot \theta} = \tan \theta \).

Step 1: Rewrite in terms of sine and cosine

Denote \( \theta \) as the given angle. Replace \( \cos \theta \), \( \sec \theta \), and \( \cot \theta \) by their respective equivalents in terms of sine and cosine, \( \frac{1}{\sin \theta} \), \( \frac{1}{\cos \theta} \), and \( \frac{\cos \theta}{\sin \theta} \). The expression hence transforms into \( \frac{\cos \theta (1/\cos \theta)}{\cos \theta/\sin \theta} \).

Step 2: Simplify the obtained expression

The first division in the expression obtained in Step 1 leads to 1 (since \( \cos \theta * 1/\cos \theta \) equals 1), while in the denominator the cosine terms cancel, leaving only \( 1/\sin \theta \). Thus, the expression simplifies to \( 1/(1/\sin \theta) \) or \( \sin \theta \).

Step 3: Convert sin to tan

Rewrite \( \sin \theta \) as \( \frac{\sin \theta}{1} \), which is equal to \( \frac{\sin \theta}{\cos \theta / \cos \theta} \), which can be further simplified as \( \tan \theta \) (since \( \tan \theta = \sin \theta / \cos \theta \))