The value of $\cos \left({\tan }^{-1}\frac{3}{4}\right)+\cos \left({\tan }^{-1}x\right)$ is equal to

• A

  $\frac{4}{5}+\frac{x}{\sqrt{1+x^2}}$
• B

  $\frac{3}{5}+\frac{1}{\sqrt{1+x^2}}$
• Correct answer
  C

  $\frac{4}{5}+\frac{1}{\sqrt{1+x^2}}$
• D

  $\frac{3}{5}+\frac{x}{\sqrt{1x^2}}$

Solution

$\cos \left({\tan }^{-1}\frac{3}{4}\right)+\cos \left({\tan }^{-1}x\right)$
$=\cos {\cos }^{-1}\left(\frac{4}{5}\right)+\cos {\cos }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)$
$=\frac{4}{5}+\frac{1}{\sqrt{1+x^2}}$

Hence, the correct answer is option: 'C'.