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Chapter 5: Problem 64

Prove the identity. $$\cos \left(\frac{5 \pi}{4}-x\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x)$$

Short Answer

Expert verified
The given identity is incorrect. The correct identity should be \(\cos \left(\frac{5 \pi}{4}-x\right)=-\frac{\sqrt{2}}{2}(\cos x - \sin x) \).

Step by step solution

01

Apply the formula for cosine of difference of angles

From the formula \(\cos (a-b) = \cos a \cos b + \sin a \sin b\), substitute \(a = \frac{5 \pi}{4}\) and \(b = x\). This gives \(\cos \left(\frac{5 \pi}{4}-x\right)= \cos \frac{5 \pi}{4} \cos x + \sin \frac{5 \pi}{4} \sin x\)
02

Substitute the values of cosine and sine

The values of \(\cos \frac{5 \pi}{4} = -\frac{\sqrt{2}}{2}\) and \(\sin \frac{5 \pi}{4} = \frac{\sqrt{2}}{2}\). Hence, the expression becomes \(-\frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x\)
03

Factor out the term \(-\frac{\sqrt{2}}{2}\)

The expression becomes \(-\frac{\sqrt{2}}{2} (\cos x - \sin x).\) But this doesn't match with the RHS of the given identity which is \(-\frac{\sqrt{2}}{2} (\cos x + \sin x).\) Therefore, the given identity is not correct.

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