Question

If sin^-1 $(\frac{1}{5})$ + sec^-1(2) + 2tan^-1$(\frac{1}{\sqrt{3}})$ + sec^-1(5) +sin^-1($\frac{1}{2}$) +2 tan^-1($\sqrt{3}$)=k$\pi$, then k =

• A. $1$
• B. $2$
• C. $4$
• D. $5$

Answer 1

Answer: (B)

The given question can be written as
$si{n}^{-1}\left(\frac{1}{5}\right)+se{c}^{-1}\left(5\right)+se{c}^1\left(2\right)+si{n}^{-1}\left(\frac{1}{2}\right)+2ta{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
$+2ta{n}^{-1}\left(\sqrt{3}\right)=k\pi ,$
or $\left\{si{n}^{-1}\left(\frac{1}{5}\right)+co{s}^{-1}\left(\frac{1}{5}\right)\right\}+\left\{{\sec }^{-1}\left(2\right)+cose{c}^{-1}\left(2\right)\right\}$
$+\left\{2ta{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)+2co{t}^{-1}\left(\frac{1}{\sqrt{3}}\right)\right\}=k\pi$
or $\frac{\pi }{2}+\frac{\pi }{2}+2\left\{ta{n}^{-1}\frac{1}{\sqrt{3}}+co{t}^{-1}\frac{1}{\sqrt{3}}\right\}=k\pi$
or $\pi +2\left(\frac{\pi }{2}\right)=k\pi$
or $\pi +\pi =k\pi$
or $2\pi =k\pi$
$k=2$