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Chapter 5: Problem 14

Verify the identity. $$\cos ^{2} \beta-\sin ^{2} \beta=2 \cos ^{2} \beta-1$$

Short Answer

Expert verified
The identity \(\cos^2\beta - \sin^2\beta = 2\cos^2\beta - 1\) holds solidly true.

Step by step solution

01

Express one of the terms in terms of the other using Pythagorean identity

Considering the Pythagorean identity \(\cos^2\beta + \sin^2\beta = 1\), express \(\sin^2\beta\) in terms of \(\cos^2\beta\). Thus, \(\sin^2\beta = 1 - \cos^2\beta\).
02

Substitute in given equation

Replace \(\sin^2\beta\) in the initially given equation \(\cos^2\beta - \sin^2\beta = 2\cos^2\beta - 1\) with the value obtained from the Pythagorean identity. Then we have, \(\cos^2\beta - (1 - \cos^2\beta) = 2\cos^2\beta - 1\) . This simplifies to \(2\cos^2\beta - 1 = 2\cos^2\beta - 1\) which proves the given trigonometric identity.
03

Final Statement

The replacement of \(\sin^2\beta\) with \(1 - \cos^2\beta\) demonstrated the left hand side of the equation equals the right hand side, thus proving the identity to be true.

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