Prove that: ^2 4 - ^2 12 = 3 4 - Vidyadip

Question

Prove that: $\cos^2\frac{\pi}{4}-\sin^2\frac{\pi}{12}=\frac{\sqrt{3}}{4}$

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Answer

$\text{L.H.S}=\cos^245^\circ-\sin^215^\circ$ $=\Big(\frac{1}{\sqrt{2}}\Big)^2-\sin^215^\circ$ $=\frac{1}{2}-\Big(\frac{1-\cos^2\times15^\circ}{2}\Big)$ $=\frac{1}{2}-\Big(\frac{1-\cos\times30^\circ}{2}\Big)$ $=\frac{1-1+\cos30^\circ}{2}$ $=\frac{\cos30^\circ}{2}$ $=\frac{\sqrt{3}}{2}\times\frac{1}{2}$ $=\frac{\sqrt{3}}{2}$ $=\text{R.H.S}$ $\therefore\text{L.H.S}=\text{R.H.S}$ Hence proved.

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