Prove that: ( 4 +x )- ( 4 -x )= 2 - Vidyadip

Question

Prove that:
$\cos\Big(\frac{\pi}{4}+\text{x}\Big)-\cos\Big(\frac{\pi}{4}-\text{x}\Big)=\sqrt2\cos\text{x}$

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Answer

We have,
$\text{LHS}=\cos\Big(\frac{\pi}{4}-\text{x}\Big)+\cos\Big(\frac{\pi}{4}+\text{x}\Big)$
$=\ 2\cos\frac{\pi}{4}\cos\text{x}$ $[\because\ \cos(\text{A}+\text{B})+\cos(\text{A}-\text{B})=2\cos\text{A}\cos\text{B}]$
$=\ 2\times\frac{1}{\sqrt2}\times\cos\text{x}$
$=\ \frac{\sqrt2\times\sqrt2}{\sqrt2}\cos\text{x}$
$=\ \sqrt2\cos\text{x}$
$=\ \text{RHS}$
$\therefore\ \cos\Big(\frac{\pi}{4}+\text{x}\Big)+\cos\Big(\frac{\pi}{4}-\text{x}\Big)=\sqrt2\cos\text{x}.$

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