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Chapter 5: Problem 10

Verify each identity. $$\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1$$

Short Answer

Expert verified
The given trigonometric identity \(\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1\) is therefore verified.

Step by step solution

01

Replace \(\sin ^{2} x\) using the Pythagorean identity

Start with the left side of the equation. \(\sin ^{2} x\) is equivalent to \(1 - \cos ^{2} x\), stemming from the Pythagorean identity \(\sin ^{2} x + \cos ^{2} x = 1\). So, we replace \(\sin ^{2} x\) with \(1 - \cos ^{2} x\), resulting in \(\cos ^{2} x - (1 - \cos ^{2} x)\).
02

Simplify the equation

Simplify the equation resulting from the previous step. \(\cos ^{2} x - (1 - \cos ^{2} x)\) simplifies to \(2 \cos ^{2} x - 1\), by distributing the minus sign to the \(1\) and the \(-\cos ^{2} x\). This resulting equation is now the same as the right side of the initial given equation.

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