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Chapter 5: Problem 61

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

Short Answer

Expert verified
The given trigonometric identity \(\sin \left(\frac{\pi}{2}-x\right)=\cos x\) is true as it follows the co-function identity of sine and cosine.

Step by step solution

01

- Understand the Co-function Identity

The co-function identity states that sine of an angle is equal to the cosine of its complementary angle. Meaning, if \(a + b = 90^o\) or \(\frac{蟺}{2}\), then \(\sin a = \cos b\) and \(\cos a = \sin b\). This is an important concept in solving the problem.
02

- Rewrite '蟺/2 - x' as a result of step 1

According to the co-function identity mentioned in the previous step, \(a + b = 90^o\) or \(a + b = \frac{\pi}{2}\). Now, \(\sin \left(\frac{\pi}{2}-x\right)\), consider it as \(a + b = \frac{\pi}{2}\), where \(a = \frac{\pi}{2}\) and \(b = -x\). Hence, \(\frac{\pi}{2} - x\) can be written as \(90^o - x\).
03

- Apply the Co-function Identity

Now if \(a = 90^o - x\) then \(b = x\), because \(a + b = 90^o\). Therefore \(b = x\). So, according to the co-function identity, \(\sin (90^o - x) = \cos x\). It implies that \(\sin \left(\frac{\pi}{2}-x\right)=\cos x\). Which means the given identity is true.

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