Problem 36 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

After evaluating the identity using the cosine of difference formula, we find that indeed \(\cos (\pi-x)=-\cos x\).

Step 1: Setup

We are asked to prove, for any angle \(x\), that \(\cos (\pi-x)=-\cos x\).

Step 2: Using the cosine of difference identity

We know that the cosine of difference of two angles is given by \(\cos(a - b) = \cos a \cos b + \sin a \sin b\). So we can rewrite our expression as \(cos(\pi - x) = \cos\pi \cos x + \sin\pi \sin x\).

Step 3: Evaluate

We know that \(\cos\pi = -1\) and \(\sin\pi = 0\). So our expression becomes \(-1 \cdot \cos x + 0 = -\cos x\).