Functions
Trigonometric Functions
JEE Main 2019
Grade 12
Question:
Let $f_k(x) = \frac{1}{k}(\sin^k x+\cos^k x)$ for $k = 1, 2, 3, \dots$ Then for all $x \in \mathbb{R}$, the value of $f_4(x)-f_6(x)$ is:
(1) $\frac{1}{4}$
(2) $\frac{1}{12}$
(3) $\frac{1}{6}$
(4) $\frac{1}{3}$
Step-by-Step Solution
Key Concept: Use the identities $\sin^4 x+\cos^4 x = 1-2 \sin^2 x \cos^2 x$ and $\sin^6 x+\cos^6 x = 1-3 \sin^2 x \cos^2 x$. Substitute and simplify the combination $f_4 - f_6$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)