Question

The value of $\cot (\frac{\pi }{4}-2{\cot }^{-1}3)$, is:

• A. $1$
• B. $7$
• C. $-1$
• D. None of these

Answer 1

Answer: (B)

$7$

Let ${\cot }^{-1}3=x$

$\Rightarrow \cot x=3$

$\Rightarrow \cot (\frac{\pi }{4}-2{\cot }^{-1}3)=\cot (\frac{\pi }{4}-2x)$

$=\frac{1}{\tan (\frac{\pi }{4}-2x)}=\frac{1+\tan 2x}{1-\tan 2x}$

Now, $\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}=\frac{2\times \frac{1}{3}}{1-{\left(\frac{1}{3}\right)}^2}=\frac{3}{4}$

Final answer $=\frac{1+\frac{3}{4}}{1-\frac{3}{4}}=7$