Question

The range of function $f\left(x\right)={\cos }^{4}x-{\sin }^{4}x$ is

• A. $[-\frac{1}{2},\frac{1}{2}]$
• B. $(-1,1)$
• C. $[-1,1]$
• D. $[1,\frac{3}{2}]$

Answer 1

The correct option is C $[-1,1]$

$f\left(x\right)={\cos }^{4}x-{\sin }^{4}x$
$=({\cos }^{2}x+{\sin }^{2}x)({\cos }^{2}x-{\sin }^{2}x)$ $\dots [\because (a+b\left)\right(a-b)=a^2-b^2]$

$=\cos 2x$ ..... $[\because {\sin }^{2}x+{\cos }^{2}x=1\text{and}{\cos }^{2}x-{\sin }^{2}x=\cos 2x]$

$\therefore f\left(x\right)=\cos 2x$

We know that, $-1\le \cos x\le 1$
$⟹\cos 2x\in [-1,1]$

Hence, range of $f\left(x\right)$ is $[-1,1]$