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Chapter 5: Problem 35

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\cot \theta \sec \theta$$

Short Answer

Expert verified
The simplified expression is \(\frac{1}{\sin (\theta)}\) or \(\csc (\theta)\)

Step by step solution

01

Use Fundamental Identities

Start by replacing each trigonometric function with its equivalent. Substitute \(\cot (\theta)\) with \(\frac{1}{\tan (\theta)}\) and \(\sec (\theta)\) with \(\frac{1}{\cos (\theta)}\). This gives: \(\frac{1}{\tan (\theta)} \frac{1}{\cos (\theta)}\)
02

Apply Trigonometric Identity

Remembering that \(\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}\), substitute this identity in the equation obtained in Step 1: \(\frac{1}{\frac{\sin (\theta)}{\cos (\theta)}} \frac{1}{\cos (\theta)}\)
03

Simplify the Equation

The fraction within the fraction can be simplified by taking the reciprocal of the denominator: \(\frac{\cos (\theta)}{\sin (\theta)} \frac{1}{\cos (\theta)}\). Further simplifications give the expression: \(\frac{1}{\sin (\theta)}\)

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