Question

Period of the function $f(x) = sin^2 2x + cos^2 2x + 2$ is

• A. $\frac{\pi}{4}$
• B. $\frac{\pi}{2}$
• C. $\frac{3\pi}{4}$
• D. $\pi$

Answer 1

Answer: (B)

$\frac{\pi}{2}$

Answer 2

The given function is $f(x) = \sin^2 2x + \cos^2 2x + 2$. Using the fundamental trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$, we can simplify the function. Here, $\theta = 2x$. So, $\sin^2 2x + \cos^2 2x = 1$. Substituting this into the expression for $f(x)$, we get: $f(x) = 1 + 2 = 3$.

The function $f(x) = 3$ is a constant function. While a constant function strictly speaking has no fundamental period, the structure of the question and the options suggest that the intended answer is related to the periods of the constituent trigonometric terms. The period of $\sin^2 2x$ is $\frac{\pi}{2}$. The period of $\cos^2 2x$ is $\frac{\pi}{2}$. The LCM of $\frac{\pi}{2}$ and $\frac{\pi}{2}$ is $\frac{\pi}{2}$.