Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Prove That: Cos π 12 − Sin π 12 = 1 √ 2 - Mathematics

Sum

Prove that:

$\cos\frac{\pi}{12} - \sin\frac{\pi}{12} = \frac{1}{\sqrt{2}}$

#### Solution

$LHS = \cos \frac{\pi}{12} - \sin \frac{\pi}{12}$
$= \cos \left( \frac{\pi}{2} - \frac{5\pi}{12} \right) - \sin \frac{\pi}{12}$
$= \sin \left( \frac{5\pi}{12} \right) - \sin \frac{\pi}{12}$
$= 2\sin \left( \frac{\frac{5\pi}{12} - \frac{\pi}{12}}{2} \right) \cos \left( \frac{\frac{5\pi}{12} + \frac{\pi}{12}}{2} \right) \left\{ \because \sin A - \sin B = 2\sin \left( \frac{A - B}{2} \right) \cos \left( \frac{A + B}{2} \right) \right\}$
$= 2\sin \left( \frac{\pi}{6} \right) \cos \left( \frac{\pi}{4} \right)$
$= 2 \times \frac{1}{2} \times \frac{1}{\sqrt{2}}$
$= \frac{1}{\sqrt{2}}$
Hence, LHS = RHS.

Concept: Transformation Formulae
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 8 Transformation formulae
Exercise 8.2 | Q 3.6 | Page 17
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