Question

$cos^4 \frac{\pi}{12} - sin^4 \frac{\pi}{12}$ is equal to

• A. $\frac1{2}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{\sqrt{3}+1}{2}$
• D. $\frac{\sqrt{3}-1}{2}$
• E. $\frac{\sqrt{2}}{2}$

Answer 1

The Correct Option is B

Given expression:

\(\cos^4 \frac{\pi}{12} - \sin^4 \frac{\pi}{12}\) 

Using the identity:

\(a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)\)

We rewrite:

\(\cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\)

Since \(\cos^2 x + \sin^2 x = 1\), we simplify to:

\(\cos^2 \frac{\pi}{12} - \sin^2 \frac{\pi}{12}\)

Using the identity:

\(\cos^2 x - \sin^2 x = \cos 2x\)

Substituting \(x = \frac{\pi}{12}\):

\(\cos^2 \frac{\pi}{12} - \sin^2 \frac{\pi}{12} = \cos \frac{\pi}{6}\)

Since \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), we get:

\(\frac{\sqrt{3}}{2}\)