Question

${\cos }^{-1}\left(\frac{15}{17}\right)+2{\tan }^{-1}\left(\frac{1}{5}\right)=$

• A. $\frac{\pi }{2}$
• B. ${\cos }^{-1}\left(\frac{171}{221}\right)$
• C. $\frac{\pi }{4}$
• D. none of these

Answer 1

The correct option is D

none of these

Explanation for correct option:

Step-1: Apply formula $2{\tan }^{-1}\left(x\right)={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$

Given, ${\cos }^{-1}\left(\frac{15}{17}\right)+2{\tan }^{-1}\left(\frac{1}{5}\right)$

$$\begin{gathered} ={\cos }^{-1}\left(\frac{15}{17}\right)+{\cos }^{-1}\left(\frac{1-{\left({\displaystyle \frac{1}{5}}\right)}^2}{1+{\left({\displaystyle \frac{1}{5}}\right)}^2}\right) \\ ={\cos }^{-1}\left(\frac{15}{17}\right)+{\cos }^{-1}\left(\frac{12}{13}\right) \end{gathered}$$

Step-2: Apply formula ${\cos }^{-1}\left(x\right)+{\cos }^{-1}\left(y\right)={\cos }^{-1}\left(xy-\sqrt{1-x^2}\times \sqrt{1-y^2}\right)$

$$\begin{gathered} ={\cos }^{-1}\left(\left(\frac{15}{17}\times \frac{12}{13}\right)-\left(\sqrt{1-{\left(\frac{15}{17}\right)}^2}\times \sqrt{1-{\left(\frac{12}{13}\right)}^2}\right)\right) \\ ={\cos }^{-1}\left(\frac{180}{221}-\left(\sqrt{\frac{289-225}{289}}\times \sqrt{\frac{169-144}{169}}\right)\right) \\ ={\cos }^{-1}\left(\frac{180}{221}-\left(\frac{8}{17}\times \frac{5}{13}\right)\right) \\ ={\cos }^{-1}\left(\frac{180}{221}-\frac{40}{221}\right) \\ ={\cos }^{-1}\left(\frac{140}{221}\right) \end{gathered}$$

Hence, correct answer is option $\left(D\right)$.