Problem 38
Question
Verify each identity. $$\tan (\pi-x)=-\tan x$$
Step-by-Step Solution
Verified Answer
\(\tan (\pi - x) = -\tan x\) is a true identity.
1Step 1: Apply the Trigonometric Identity for Tangent
Start by applying the definition of the tangent function. This means replacing \(\tan (x)\) with \(\sin (x) / \cos (x)\). So, \(\tan (\pi - x) = \sin (\pi - x) / \cos (\pi - x)\) and \(-\tan (x) = -\sin (x) / \cos (x)\). This is going to be the foundation of the solution.
2Step 2: Use the Properties of Sine and Cosine
Now, using the properties of sine and cosine, we have \(\sin (\pi - x) = \sin \pi \cos x - \cos \pi \sin x\), which simplifies to \(-\sin x\), and \(\cos (\pi - x) = \cos \pi \cos x + \sin \pi \sin x\), which simplifies to \(-\cos x\). So, now our equation is \(-\sin x / -\cos x\).
3Step 3: Simplify the Expression
Simplify the equation by dividing both terms by \(-1\). We will be left with \(\sin x / \cos x\), which is equal to \(\tan x\). This verifies that \(\tan (\pi - x) = -\tan x\).
Other exercises in this chapter
Problem 38
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