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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
Verification completed successfully, the identity \(\cos^2 \beta - \sin^2 \beta = 2 \cos^2 \beta -1\) is true.

Step by step solution

01

Write the equation and recognize the identities to use

We are given the following equation to verify: \(\cos^2 \beta - \sin^2 \beta = 2 \cos^2 \beta -1\). Here we can identify that left side is a double-angle formula for cosine: \(\cos (2\beta) = \cos^2 \beta - \sin^2 \beta\), while on the right side, we can rearrange this as \(\cos^2 \beta = (2 \cos^2 \beta - 1) / 2\). This looks like a digressing formula: \(\cos^2 \beta = (1+ \cos(2 \beta))/2\)
02

Rewrite using the double-angle formula

Rewrite the left-hand side of the equation using the identity \(\cos (2\beta) = \cos^2 \beta - \sin^2 \beta\). This gives us: \(\cos (2\beta) = 2 \cos^2 \beta -1\)
03

Rearrange the right-hand side

Rewrite the right-hand side of the equation to look like a digressing formula, \(\cos^2 \beta = (1+ \cos(2 \beta))/2\). Therefore, \((1+ \cos(2 \beta))/2 = 2 \cos^2 \beta -1\)
04

Comparison

From steps 2 and 3, we see that both sides of the equation are equivalent, hence the identity is verified.

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