Question

The value of $\tan \left[{\cos }^{-1}\left(-\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]$ is

• A. $\frac{2}{3\sqrt{5}}$
• B. $\frac{2}{3}$
• C. $\frac{1}{\sqrt{5}}$
• D. $\frac{4}{\sqrt{5}}$

Answer 1

The correct option is A

$\frac{2}{3\sqrt{5}}$

The explanation for the correct option

The given trigonometric expression: $\tan \left[{\cos }^{-1}\left(-\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]$.

$$\begin{gathered} \tan \left[{\cos }^{-1}\left(-\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]=\tan \left[\mathrm{\pi }-{\cos }^{-1}\left(\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]\left[\because {\cos }^{-1}\left(-\theta \right)=\mathrm{\pi }-{\cos }^{-1}\left(\theta \right)\right] \\ \Rightarrow \tan \left[{\cos }^{-1}\left(-\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]=\tan \left[\frac{\mathrm{\pi }}{2}-{\cos }^{-1}\left(\frac{2}{7}\right)\right] \\ \tan \left[{\cos }^{-1}\left(-\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]=\tan \left[{\sin }^{-1}\left(\frac{2}{7}\right)\right]\left[\because {\sin }^{-1}\left(\theta \right)+{\cos }^{-1}\left(\theta \right)=\frac{\mathrm{\pi }}{2}\right] \end{gathered}$$

Let us assume that ${\sin }^{-1}\left(\frac{2}{7}\right)=y$

$$\begin{gathered} \Rightarrow \sin \left(y\right)=\frac{2}{7} \\ \Rightarrow {\sin }^2\left(y\right)={\left(\frac{2}{7}\right)}^2 \\ \Rightarrow 1-{\cos }^2\left(y\right)=\frac{4}{49} \\ \Rightarrow {\cos }^2\left(y\right)=1-\frac{4}{49} \\ \Rightarrow {\cos }^2\left(y\right)=\frac{45}{49} \\ \Rightarrow \cos \left(y\right)=\sqrt{\frac{45}{49}} \\ \Rightarrow \cos \left(y\right)=\frac{3\sqrt{5}}{7} \end{gathered}$$

Thus, $\tan \left(y\right)=\frac{\sin \left(y\right)}{\cos \left(y\right)}$

$$\begin{gathered} \Rightarrow \tan \left(y\right)=\frac{{\displaystyle \frac{2}{7}}}{{\displaystyle \frac{3\sqrt{5}}{7}}} \\ \Rightarrow \tan \left(y\right)=\frac{2}{3\sqrt{5}} \\ \Rightarrow \tan \left[{\sin }^{-1}\left(\frac{2}{7}\right)\right]=\frac{2}{3\sqrt{5}}\left[\because {\sin }^{-1}\left(\frac{2}{7}\right)=y\right] \end{gathered}$$

Therefore, $\tan \left[{\cos }^{-1}\left(-\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]=\frac{2}{3\sqrt{5}}$.

Hence, option A is correct.