Question

Let $f\left(x\right)=\cos 2x.\cot \left(\frac{\pi }{4}-x\right)$ If $f$ is continuous at $x=\frac{\pi }{4}$ then the value of $f(\frac{\pi }{4})$ is equal to

• A. $2$
• B. $-2$
• C. $\frac{-1}{2}$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $2$

$f\left(x\right)=cos2x.cot(\frac{\pi }{4}-x)$
Given $f\left(x\right)$ is continuous at $x=\frac{\pi }{4}$
$\underset{x\to \frac{\pi }{4}}{lim}f\left(x\right)=f\left(\frac{\pi }{4}\right)$
$=\underset{x\to \frac{\pi }{4}}{lim}\frac{\cos 2x}{\tan (\frac{\pi }{4}-x)}$
It is of the form $\frac{0}{0}$ , so applying L-Hospital's rule
$=\underset{x\to \frac{\pi }{4}}{lim}\frac{-2\sin 2x}{-{\sec }^2(\frac{\pi }{4}-x)}$
$=2$