Question

If $-\frac{\mathrm{\pi }}{2}<\theta <\frac{\mathrm{\pi }}{2}$ and $\theta \ne \frac{\mathrm{\pi }}{4}$, then the value of $cot\left(\frac{\mathrm{\pi }}{4}+\theta \right)cot\left(\frac{\mathrm{\pi }}{4}-\theta \right)$ is

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is C

$1$

Explanation for the correct option:

Step-1: Apply the formulas

$cot\theta =\frac{1}{\tan \theta }$ and $\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$

Given condition $-\frac{\mathrm{\pi }}{2}<\theta <\frac{\mathrm{\pi }}{2}$and $\theta \ne \frac{\mathrm{\pi }}{4}$

Consider $cot\left(\frac{\mathrm{\pi }}{4}+\theta \right)cot\left(\frac{\mathrm{\pi }}{4}-\theta \right)$

Step 2: Find the value of $cot\left(\frac{\mathrm{\pi }}{4}+\theta \right)cot\left(\frac{\mathrm{\pi }}{4}-\theta \right)$

$\begin{array}{rcl}\therefore cot\left(\frac{\mathrm{\pi }}{4}+\theta \right)cot\left(\frac{\mathrm{\pi }}{4}-\theta \right)& =& \left(\frac{1}{\tan \left(\frac{\mathrm{\pi }}{4}+\theta \right)}\right)\left(\frac{1}{\tan \left(\frac{\mathrm{\pi }}{4}-\theta \right)}\right);\left[\therefore \cot \left(\mathrm{\theta }\right)=\frac{1}{\tan \theta }\right]\\ & =& \left(\frac{1}{\left({\displaystyle \frac{\tan \frac{\mathrm{\pi }}{4}+\tan \theta }{1-\tan \frac{\mathrm{\pi }}{4}\tan \theta }}\right)}\right)\left(\frac{1}{\left({\displaystyle \frac{\tan \frac{\mathrm{\pi }}{4}-\tan \theta }{1+\tan \frac{\mathrm{\pi }}{4}\tan \theta }}\right)}\right);\left[\because \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}\right]\\ & =& \left(\frac{1-\tan \frac{\mathrm{\pi }}{4}\tan \theta }{\tan \frac{\mathrm{\pi }}{4}+\tan \theta }\right)\left(\frac{1+\tan \frac{\mathrm{\pi }}{4}\tan \theta }{\tan \frac{\mathrm{\pi }}{4}-\tan \theta }\right)\end{array}$

we know $\tan \frac{\mathrm{\pi }}{4}=1$

$\begin{array}{rcl}\therefore cot\left(\frac{\mathrm{\pi }}{4}+\theta \right)cot\left(\frac{\mathrm{\pi }}{4}-\theta \right)& =& \left(\frac{1-\tan \theta }{1+\tan \theta }\right)\left(\frac{1+\tan \theta }{1-\tan \theta }\right)\\ & =& 1\end{array}$

Therefore, the value of $cot\left(\frac{\mathrm{\pi }}{4}+\theta \right)cot\left(\frac{\mathrm{\pi }}{4}-\theta \right)$ = $1$

Hence, the correct answer is option (C)