Question

The value of $\sin \left[{\sin }^{-1}\left(\frac{1}{3}\right)+{\sec }^{-1}\left(3\right)\right]+\cos \left[{\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(2\right)\right]$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A

$1$

Explanation for correct option:
Given expression is $\sin \left[{\sin }^{-1}\left(\frac{1}{3}\right)+{\sec }^{-1}\left(3\right)\right]+\cos \left[{\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(2\right)\right]$
$$\begin{gathered} {\sec }^{-1}\left(3\right)={\cos }^{-1}\left(\frac{1}{3}\right)\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }{\mathbf{sec}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{x}}\mathbf{\right)}\mathbf{]} \\ {\tan }^{-1}\left(2\right)={\cot }^{-1}\left(\frac{1}{2}\right)\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{cot}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{x}}\mathbf{\right)}\mathbf{]} \end{gathered}$$
$$\begin{gathered} \Rightarrow \sin \left[{\sin }^{-1}\left(\frac{1}{3}\right)+{\sec }^{-1}\left(3\right)\right]+\cos \left[{\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(2\right)\right] \\ =\sin \left[{\sin }^{-1}\left(\frac{1}{3}\right)+{\cos }^{-1}\left(\frac{1}{3}\right)\right]+\cos \left[{\tan }^{-1}\left(\frac{1}{2}\right)+{\cot }^{-1}\left(\frac{1}{2}\right)\right] \end{gathered}$$
$$\begin{gathered} =\sin \left(\frac{\mathrm{\pi }}{2}\right)+\cos \left(\frac{\mathrm{\pi }}{2}\right)\mathbf{\left[}\mathbf{\because }{\mathbf{sin}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{}\mathbf{\&}\mathbf{}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}{\mathbf{cot}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{\right]} \\ =1+0\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{sin}\mathbf{\left(}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{\&}\mathbf{}\mathbf{cos}\mathbf{\left(}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{]} \\ =1 \end{gathered}$$

Hence option(A) is correct