Question

Let $f_k\left(x\right)=\frac{1}{k}(si{n}^{k}x+co{s}^{k}x)$ for $k=1,2,3,....$ Then for all $x\in R$, the value of $f_4\left(x\right)-f_6\left(x\right)$ is equal to:-

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

Answer 1

Answer: (D)

$\frac{1}{12}$

$f_4\left(x\right)-f_6\left(x\right)$

$=\frac{1}{4}(si{n}^{4}x+co{s}^{4}x)-\frac{1}{6}(si{n}^{6}x+co{s}^{6}x)$

$=\frac{1}{4}(1-\frac{1}{2}si{n}^{2}2x)-\frac{1}{6}(1-\frac{3}{4}si{n}^{2}2x)=\frac{1}{12}$