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Chapter 5: Problem 3

Verify each identity. $$\tan (-x) \cos x=-\sin x$$

Short Answer

Expert verified
The given identity \(\tan (-x) \cos x = -\sin x\) is true.

Step by step solution

01

Define the Trigonometric Identity for Tan

The trigonometric function \(\tan(x)\) can be defined as \(\sin(x) / \cos(x)\). So, \(\tan (-x) = \sin (-x) / \cos (-x)\).
02

Simplify the Expression

Substitute \(\tan (-x)\) in terms of \(\sin (-x)\) and \(\cos (-x)\) to get \(\tan (-x) \cos x = \sin (-x) / \cos (-x) * \cos x\). Simplifying this, we get \(\sin (-x)\).
03

Use the Property of Sine

The sine function is odd, which means \(\sin (-x) = - \sin x\). Substituting this into our expression, we get \(- \sin x\).

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