Question

The period of \( \sin(4x + \cos^{4} x) \) is

• A. $\frac{\pi^{4}}{2}$
• B. $\frac{\pi^{2}}{2}$
• C. $\frac{\pi}{4}$
• D. $\frac{\pi}{2}$

Hint:

Period of $\sin^n x + \cos^n x$ is $\pi/2$ when $n$ is even.

Answer 1

The Correct Option is D

Step 1: Simplify expression
$\sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2 \sin^2 x \cos^2 x$.
Step 2: Use double angle identities
$= 1 - \frac{1}{2}(2 \sin x \cos x)^2 = 1 - \frac{1}{2} \sin^2(2x)$.
Step 3: Further reduction
$= 1 - \frac{1}{4}(1 - \cos 4x) = \frac{3}{4} + \frac{\cos 4x}{4}$.
Step 4: Determine period
The period of $\cos(kx)$ is $\frac{2\pi}{k}$. For $k=4$, the period is $\frac{2\pi}{4} = \frac{\pi}{2}$.
Final Answer: (D)