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Question
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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