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Chapter 6: Problem 75

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\cos x \cot x+\sin x$$

Short Answer

Expert verified
The given expression simplifies to the cosecant function, \( \csc x \) .

Step by step solution

01

Simplify the given expression

The given expression is \( \cos x \cot x+ \sin x \). The cotangent function can be expressed in terms of sine and cosine functions as \( \cot x = \frac{\cos x}{\sin x} \). Substituting \( \cot x \) in the given expression with this identity, it becomes \( \cos x \cdot \frac{\cos x}{\sin x} + \sin x \) which simplifies to \( \frac{\cos^2 x}{\sin x} + \sin x \)
02

Make use of trigonometric identities

Now, keep in mind the Pythagorean identity in trigonometry: \( \sin^2 x + \cos^2 x = 1 \). From this identity, we can express \( \cos^2 x \) as \( 1 - \sin^2 x \). Replace \( \cos^2 x \) in the expression obtained in Step 1 with \( 1 - \sin^2 x \). This transforms the expression to \( \frac{1 - \sin^2 x}{\sin x} + \sin x \)
03

Simplify further

Further simplify the expression by breaking down the fraction. This gives \( \frac{1}{\sin x} - \sin x + \sin x \). The positive and negative \( \sin x \) cancel each other out, leaving us with \( \frac{1}{\sin x} \)
04

Identify the resulting function

Lastly, you should recognize \( \frac{1}{\sin x} \) as the definition of the function \( \csc x \) (cosecant). Thus, the original expression simplifies to cotangent function.

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Most popular questions from this chapter

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\csc \frac{u}{2}=\pm \sqrt{\frac{2 \csc u}{\csc u-\cot u}}$$

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