Prove that cos left( frac{pi}{4} + x right) + | vedclass

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We use the trigonometric identity $\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)$.Let $A = \frac{\pi}{4} + x

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We use the trigonometric identity $\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)$.
Let $A = \frac{\pi}{4} + x$ and $B = \frac{\pi}{4} - x$.
Then,$\frac{A+B}{2} = \frac{(\frac{\pi}{4} + x) + (\frac{\pi}{4} - x)}{2} = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4}$.
And,$\frac{A-B}{2} = \frac{(\frac{\pi}{4} + x) - (\frac{\pi}{4} - x)}{2} = \frac{2x}{2} = x$.
Substituting these into the identity:
$L.H.S. = 2 \cos \left( \frac{\pi}{4} \right) \cos x$.
Since $\cos \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}}$,we have:
$L.H.S. = 2 \times \frac{1}{\sqrt{2}} \cos x = \sqrt{2} \cos x = R.H.S$.

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

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