Let f_k(x) = 1/k(sin^k x + cos^k x) for k = 1, 2, 3 … Then, for all x ∈ R, the value of f_4(x) - f_6(x) is equal to

Let f_k(x) = 1/k(sin^k x + cos^k x) for k = 1, 2, 3 … Then,… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 1 Jan · Shift I

Let

f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)

for

k = 1, 2, 3 \ldots

Then, for all

x \in R

, the value of

f_4(x) - f_6(x)

is equal to

A.
$\frac{1}{12}$
B.
$\frac{5}{12}$
C.
$-\frac{1}{12}$
D.
$\frac{1}{4}$

Show answer and solution

Correct option: A

Solution

f4(x) - f6(x) = (1/4)(sin⁴x + cos⁴x) - (1/6)(sin⁶x + cos⁶x).

Using identities: sin⁴x + cos⁴x = 1 - 2sin²x·cos²x, and sin⁶x + cos⁶x = 1 - 3sin²x·cos²x.

So f4 - f6 = (1/4)(1 - 2sin²x·cos²x) - (1/6)(1 - 3sin²x·cos²x) = 1/4 - (1/2)sin²x·cos²x - 1/6 + (1/2)sin²x·cos²x = 1/4 - 1/6 = 1/12.

[JEE Main 2019 · 1 Jan · Shift I]

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