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: $\frac{2}{\sqrt{3}}$
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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