Question : Find the value of $\frac{\cos^2 15^{\circ}-\sin^2 15^{\circ}}{\cos^2 145^{\circ}+\sin^2 145^{\circ}}$.
Option 1: $\frac{1}{\sqrt{3}}$
Option 2: $\frac{1}{1-\sqrt{3}}$
Option 3: $\frac{\sqrt{3}}{2}$
Option 4: $\frac{2}{\sqrt{3}}$
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Correct Answer: $\frac{\sqrt{3}}{2}$
Solution : $\frac{\cos^2 15^{\circ}-\sin^2 15^{\circ}}{\cos^2 145^{\circ}+\sin^2 145^{\circ}}$ = $\frac{\cos^2 15^{\circ}-\sin^2 15^{\circ}}{1}$ = $\cos (2×15)^{\circ}$ (Since $\cos^2A-\sin^2A=\cos2A$) = $\cos 30^{\circ}$ = $\frac{\sqrt 3}2$ Hence, the correct answer is $\frac{\sqrt 3}2$.
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