Prove That: Cos 2 45 ∘ − Sin 2 15 ∘ = √ 3 4 - Mathematics

Prove that:
$\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}$

Solution

$\cos^2 45^\circ - \sin^2 15^\circ$
$= \cos\left( 45^\circ + 15^\circ \right)\cos\left( 45^\circ - 15^\circ \right) \left[ \cos^2 X - \sin^2 Y = \cos\left( X + Y \right)\cos\left( X - Y \right) \right]$
$= \cos60^\circ\cos30^\circ$
$= \frac{1}{2} \times \frac{\sqrt{3}}{2}$
$= \frac{\sqrt{3}}{4}$
Hence proved.

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 7 Values of Trigonometric function at sum or difference of angles
Exercise 7.1 | Q 15.1 | Page 20