Question

If $f_n\left(x\right)=\frac{1}{n}({\cos }^{n}x+{\sin }^{n}x),$ for $n=1,2,3,....,$ then $f_4\left(x\right)-f_6\left(x\right)$ is equal to

• A. $10$
• B. $\frac{1}{12}$
• C. $\frac{1}{10}$
• D. $12$

Answer 1

The correct option is B $\frac{1}{12}$

Let $\sin x=s,\cos x=c$
$f_4\left(x\right)-f_6\left(x\right)=\frac{1}{4}(s^4+c^4)-\frac{1}{6}(s^6+c^6)$

$s^4+c^4=1-2s^2{c}^2{s}^6+c^6=1-3s^2{c}^2$

$\Rightarrow f_4\left(x\right)-f_6\left(x\right)=\frac{1}{4}-\frac{1}{6}=\frac{1}{12}$