Question

Let $f\left(x\right)={\sin }^{2}x+{\cos }^{4}x+2$ and $g\left(x\right)=\cos \left(\cos x\right)+\cos \left(\sin x\right)$. Also let period of $f\left(x\right)$ and $g\left(x\right)$ be $T_1$ and $T_2$ respectively then

• A. $T_1=2T_2$
• B. $2T_1=T_2$
• C. $T_1=T_2$
• D. $T_1=4T_2$

Answer 1

The correct option is C $T_1=T_2$

$f\left(x\right)={\sin }^2\left(x\right)+{\cos }^4\left(x\right)+2$
$f\left(x\right)={\sin }^2\left(x\right)+(1-{\sin }^2(x\left)\right)\left({\cos }^2\right(x\left)\right)+2$
$={\sin }^2\left(x\right)+{\cos }^2\left(x\right)-\frac{{\sin }^{2}2x}{4}+2$
$f\left(x\right)=3-\frac{{\sin }^{2}2x}{4}$
Hence period is $\frac{\pi }{2}$
Let $g\left(x\right)=\cos \left(\sin \right(x\left)\right)+\cos \left(\cos \right(x\left)\right)$
Hence period will be $\frac{\pi }{2}$
Therefore $T_1=T_2$