Question

$\cos^4(\pi/8) + \cos^4(3\pi/8) + \cos^4(5\pi/8) + \cos^4(7\pi/8) = \dots$

• A. $1/2$
• B. $3/2$
• C. $1/4$
• D. $3/4$

Hint:

The algebraic reductions $\sin^4 x + \cos^4 x = 1 - \frac{1}{2}\sin^2(2x)$ and $\sin^6 x + \cos^6 x = 1 - \frac{3}{4}\sin^2(2x)$ are extremely useful time-savers for evaluating high-power trigonometric series.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Use the properties $\cos(\pi - \theta) = -\cos \theta$ and $\cos(\pi/2 - \theta) = \sin \theta$.

Step 2: Formula Application:
$\cos(7\pi/8) = -\cos(\pi/8) \implies \cos^4(7\pi/8) = \cos^4(\pi/8)$. $\cos(5\pi/8) = -\cos(3\pi/8) \implies \cos^4(5\pi/8) = \cos^4(3\pi/8)$. The expression becomes $2[\cos^4(\pi/8) + \cos^4(3\pi/8)]$.

Step 3: Explanation:
Also, $\cos(3\pi/8) = \sin(\pi/8)$. Expression $= 2[\cos^4(\pi/8) + \sin^4(\pi/8)] = 2[1 - 2\sin^2(\pi/8)\cos^2(\pi/8)]$ $= 2[1 - \frac{1}{2}\sin^2(\pi/4)] = 2[1 - \frac{1}{2}(\frac{1}{2})] = 2[1 - 1/4] = 2(3/4) = 3/2$.

Step 4: Final Answer:
The value is $3/2$.