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Chapter 5: Problem 51

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\cos \left(\frac{\pi}{2}-x\right) \sec x$$

Short Answer

Expert verified
The simplified expression is \( \tan(x) \)

Step by step solution

01

Identify the Co-function Identity

The co-function identity states that the cosine of the complement of an angle is equal to the sine of the angle itself. In other words, \(\cos(\frac{\pi}{2} - x) = \sin(x)\). Replace \(\cos(\frac{\pi}{2} - x)\) in our given expression with \(\sin(x)\). After doing this, our expression becomes \(\sin(x) \sec(x)\)
02

Recall the Reciprocal Identity of secant

The reciprocal identity states that the secant of an angle is the reciprocal of the cosine of the angle. That means, \(\sec(x) = \frac{1}{\cos(x)}\). Replace \(\sec(x)\) in our updated expression with \(\frac{1}{\cos(x)}\). Therefore, our expression now turns into \(\sin(x) \cdot \frac{1}{\cos(x)} = \frac{\sin(x)}{\cos(x)}\)
03

Recognize the Quotient Identity

According to quotient identity, \(\frac{\sin(x)}{\cos(x)}\) is equal to \(\tan(x)\). So, replace the \(\frac{\sin(x)}{\cos(x)}\) in our updated expression with \(\tan(x)\).

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