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Chapter 5: Problem 14

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

Short Answer

Expert verified
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 by step solution

01

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 \( \frac{\cos \theta (1/\cos \theta)}{\cos \theta/\sin \theta} \).
02

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 \( \sin \theta \).
03

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 \( \tan \theta = \sin \theta / \cos \theta \))

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