Question

Evaluate: $\cos \left[2{\tan }^{-1}\left[\frac{1}{7}\right]\right]$

• A. $\sin \left(4{\cot }^{-1}3\right)$
• B. $\sin \left(3{\cot }^{-1}4\right)$
• C. $\cos \left(3{\cot }^{-1}4\right)$
• D. $\cos \left(4{\cot }^{-1}4\right)$

Answer 1

The correct option is A $\sin \left(4{\cot }^{-1}3\right)$

Let ${\tan }^{-1}\left[\frac{1}{7}\right]=\theta$
$\Rightarrow tan\theta =\frac{1}{7}$
$\Rightarrow cos\theta =\frac{7}{\sqrt{50}}$
We need $cos2\theta$, which is $2co{s}^2\theta -1.$
$\therefore cos2\theta =2\times \frac{49}{50}-1=\frac{24}{25}$
Now, in the first option, let $co{t}^{-1}3$ be $\alpha .$
$\Rightarrow sin\alpha =\frac{1}{\sqrt{10}}$ and $cos\alpha =\frac{3}{\sqrt{10}}$
$sin4\alpha =2sin2\alpha cos2\alpha$
$sin4\alpha =2\times 2\times \frac{1}{\sqrt{10}}\times \frac{3}{\sqrt{10}}\times (2\times \frac{9}{10}-1)$
$sin4\alpha =\frac{12}{10}\times \frac{8}{10}=\frac{96}{100}=\frac{24}{25}$