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Chapter 5: Problem 62

Prove the identity. $$\sin \left(\frac{\pi}{2}+x\right)=\cos x$$

Short Answer

Expert verified
The identity \(\sin \left(\frac{\pi}{2}+x\right)=\cos x\) has been proven to be true.

Step by step solution

01

Write down the addition formula for sine

The formula for sine of addition of two angles is: \[\sin(A + B) = \sin A \cos B + \cos A \sin B\]
02

Substitute the given values and simplify

Substitute \(A = \frac{\pi}{2}\) and \(B = x\) in the formula: \[\sin \left(\frac{\pi}{2} + x\right) = \sin \frac{\pi}{2} \cos x + \cos \frac{\pi}{2} \sin x\]Now use the fact that \(\sin \frac{\pi}{2}\) equals 1 and \(\cos \frac{\pi}{2}\) equals 0 to simplify:\[\sin \left(\frac{\pi}{2} + x\right) = 1 \cdot \cos x + 0 \cdot \sin x= \cos x\]

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