Question

Evaluate \[ \cos^4 \frac{\pi}{24} - \sin^4 \frac{\pi}{24}. \]

• A. \(\frac{\sqrt{2}-\sqrt{3}}{2}\)
• B. \(\frac{\sqrt{2}+\sqrt{3}}{2}\)
• C. \(\frac{\sqrt{2}-\sqrt{6}}{4}\)
• D. \(\frac{\sqrt{2}+\sqrt{6}}{4}\)

Hint:

Use \(\cos^4 \theta - \sin^4 \theta = \cos 2\theta\) and express angles in sums/differences of known angles for exact trigonometric values.

Answer 1

The Correct Option is D

Step 1: Reduce the expression using a double angle identity.
Since \(\cos^4\theta - \sin^4\theta = (\cos^2\theta-\sin^2\theta)(\cos^2\theta+\sin^2\theta) = \cos 2\theta\), putting \(\theta = \frac{\pi}{24}\) turns the whole expression into a single cosine, \(\cos^4\frac{\pi}{24} - \sin^4\frac{\pi}{24} = \cos\frac{\pi}{12}\).

Step 2: Split the angle using a different pair of standard angles.
Instead of writing \(\frac{\pi}{12}\) as \(45^\circ-30^\circ\), it is just as easy to write \(15^\circ = 60^\circ - 45^\circ\), so
\[ \cos 15^\circ = \cos(60^\circ-45^\circ) = \cos60^\circ\cos45^\circ + \sin60^\circ\sin45^\circ \]

Step 3: Plug in the known values.
\[ \cos 15^\circ = \frac{1}{2}\cdot\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \]

Step 4: Combine into a single fraction.
Adding the two terms gives the final value directly.
\[ \boxed{\frac{\sqrt{2}+\sqrt{6}}{4}} \]