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Chapter 5: Problem 36

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\tan (-x) \cos x$$

Short Answer

Expert verified
The simplified expression is \( -\sin x \).

Step by step solution

01

Expand tan using fundamental identities

Firstly, let's rewrite \( \tan (-x) \) using the identity \( \tan x = \frac{\sin x}{\cos x} \). This gives us \( \tan (-x) = \frac{\sin (-x)}{\cos (-x)} \). Now the expression becomes \( \frac{\sin (-x)}{\cos (-x)} \cos x \).
02

Simplify using property of sine function

Next, we use the property of sine being an odd function. Hence \( \sin (-x) = -\sin x \). Now the expression becomes \( \frac{-\sin x}{\cos -x} \cos x \).
03

Simplify expression using cosine property

We know that the cosine function is even, which means \( \cos(-x) = \cos(x) \). Applying this to our expression gives us \( \frac{-\sin x}{\cos x} \cos x \).
04

Cancel common terms

Finally, we notice that the \(\cos x\) in the denominator cancels out the \(\cos x\) in the expression, hence leaving us with \(-\sin x\) as a result.

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