Prove that: 12 - 12 =1 2 - Vidyadip

Question

Prove that:
$\cos\frac{\pi}{12}-\sin\frac{\pi}{12}=\frac{1}{\sqrt2}$

✓

Answer

$\text{LHS}=\cos\frac{\pi}{12}-\sin\frac{\pi}{12}$
Multiplying and dividing by $\sqrt2$ on LHS
$=\ \sqrt2\Big(\frac{1}{\sqrt2}\cos\frac{\pi}{12}-\frac{1}{\sqrt2}\sin\frac{\pi}{12}\Big)$
$=\ \sqrt2\Big(\sin\frac{\pi}{4}\cos\frac{\pi}{12}-\cos\frac{\pi}{4}\sin\frac{\pi}{12}\Big)$ $\Big[\because\ \frac{1}{\sqrt2}=\cos\frac{\pi}{4}=\sin\frac{\pi}{4}\Big]$
$=\ \sqrt2\Big(\sin\Big(\frac{\pi}{4}-\frac{\pi}{12}\Big)\Big)$ $[\because\ \sin(\text{A}-\text{B})=\sin\text{A}\cos\text{B}-\cos\text{A}\sin\text{B}$
$=\ \sqrt2\Big(\sin\frac{\pi}{6}\Big)$
$=\ \sqrt2\times\frac{1}{2}$
$=\ \frac{1}{\sqrt2}$
$=\ \text{RHS}$

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