Question

The value of $\cos \left(2{\tan }^{-1}\left(-7\right)\right)$ is

• A. $\frac{49}{50}$
• B. $-\frac{49}{50}$
• C. $\frac{24}{25}$
• D. $-\frac{24}{25}$
• E. $\frac{48}{49}$

Answer 1

The correct option is D

$-\frac{24}{25}$

Explanation for correct option

Given trigonometric function is $\cos \left(2{\tan }^{-1}\left(-7\right)\right)$

We know that $2{\tan }^{-1}\left(x\right)={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$

Therefore,

$\cos \left(2{\tan }^{-1}\left(x\right)\right)=\cos \left({\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right)=\frac{1-x^2}{1+x^2}\left[\because \cos \left({\cos }^{-1}\left(\theta \right)\right)=\theta \right]$

Now,

$\cos \left(2{\tan }^{-1}\left(-7\right)\right)=\cos \left({\cos }^{-1}\left(\frac{1-7^2}{1+7^2}\right)\right)=\frac{-48}{50}=-\frac{24}{25}$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{D}\right)$