Question

If $f\left(x\right)=\left(\frac{\cos 3x}{\cos x}\right),\forall x\ne (2n\pm 1)\frac{\pi }{2}$, then the range of $f\left(x\right)\forall x\in R$ is

• A. $[-3,1]$
• B. $[-\infty ,1]$
• C. $(-\infty ,1)$
• D. $(-3.-,1]$

Answer 1

Answer: (D)

$(-3.-,1]$

Let

$y=f\left(x\right)=\frac{\cos 3x}{\cos x}$

$\Rightarrow y=\frac{4{\cos }^{3}x-3\cos x}{\cos x}$

$\Rightarrow y=4{\cos }^{2}x-3$

$\Rightarrow y+3=4{\cos }^{2}x$

$\Rightarrow \frac{y+3}{4}={\cos }^{2}x$
Since, $0<{\cos }^{2}x\le 1$ $(\because {\cos }^{2}x\ne 0$ as y is not defined for $\cos x=0$)
$\Rightarrow 0<\frac{y+3}{4}\le 1$
$\Rightarrow -3<y\le 1$
Hence, range of $f\left(x\right)$ is $(-3,1]$