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Chapter 5: Problem 43

Verify the identity. $$\cos ^{2} \beta+\cos ^{2}\left(\frac{\pi}{2}-\beta\right)=1$$

Short Answer

Expert verified
Yes, the identity \( \cos ^{2} \beta + \cos ^{2}\left(\frac{\pi}{2} - \beta\right) = 1 \) is true. The key is to use the Pythagorean identity and complementary angle theorem.

Step by step solution

01

Write down the original equation

The original equation given is: \( \cos ^{2} \beta + \cos ^{2}\left(\frac{\pi}{2} - \beta\right) = 1 \)
02

Use complementary angle theorem

Using the complementary angle theorem which states that \( \cos(\frac{\pi}{2} - \beta) = \sin(\beta) \), we can rewrite the equation as: \( \cos^2 \beta + \sin^2 \beta \)
03

Apply the Pythagorean identity

The Pythagorean identity states that \( \sin^2(x) + \cos^2(x) = 1 \). Therefore, the equation simplifies to: \( 1 = 1 \)

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