Prove that: cos²(π/4) − sin²(π/12) = √3/4

Question

Prove that:

\[ \cos^2\left(\frac{\pi}{4}\right) – \sin^2\left(\frac{\pi}{12}\right) = \frac{\sqrt{3}}{4} \]

Proof

L.H.S.

\[ = \cos^2\left(\frac{\pi}{4}\right) – \sin^2\left(\frac{\pi}{12}\right) \]

\[ = \left(\cos\frac{\pi}{4}\right)^2 – \left(\sin\frac{\pi}{12}\right)^2 \]

\[ = \left(\frac{1}{\sqrt{2}}\right)^2 – \left(\sin15^\circ\right)^2 \]

\[ = \frac{1}{2} – \left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)^2 \]

\[ = \frac{1}{2} – \frac{(\sqrt{3}-1)^2}{8} \]

\[ = \frac{1}{2} – \frac{3+1-2\sqrt{3}}{8} \]

\[ = \frac{1}{2} – \frac{4-2\sqrt{3}}{8} \]

\[ = \frac{4}{8} – \frac{4-2\sqrt{3}}{8} \]

\[ = \frac{4-4+2\sqrt{3}}{8} \]

\[ = \frac{2\sqrt{3}}{8} \]

\[ = \frac{\sqrt{3}}{4} \]

R.H.S.

\[ = \frac{\sqrt{3}}{4} \]

Hence,

\[ \cos^2\left(\frac{\pi}{4}\right) – \sin^2\left(\frac{\pi}{12}\right) = \frac{\sqrt{3}}{4} \]

Hence proved.