Question

$2{\sin }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{24}{25}\right)=$

• A. $\frac{\mathrm{\pi }}{2}$
• B. $\frac{2\mathrm{\pi }}{3}$
• C. $\frac{5\mathrm{\pi }}{3}$
• D. $Noneofthese$

Answer 1

The correct option is A

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct option:

Step 1. Solve the given expression:

$2{\sin }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{24}{25}\right)$

$={\sin }^{-1}\left[2\times \frac{3}{5}\times \sqrt{1-{\left(\frac{3}{5}\right)}^2}\right]+{\cos }^{-1}\frac{24}{25}$ $\left[\mathbf{\because }\mathbf{2}\mathbf{}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathit{x}\mathbf{=}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{2}\mathit{x}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{\right)}\right]$

$={\sin }^{-1}\left(\frac{24}{25}\right)+{\cos }^{-1}\left(\frac{24}{25}\right)$

$={\cos }^{-1}{\sqrt{1-\left(\frac{24}{25}\right)}}^2+{\cos }^{-1}\frac{24}{25}$ $\left[\mathbf{\because }\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathit{x}\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{\right)}\right]$

$={\cos }^{-1}\left(\frac{7}{25}\right)+{\cos }^{-1}\left(\frac{24}{25}\right)$

Step 2. Apply formula $\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\mathit{x}\mathbf{-}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\mathit{y}\mathbf{=}\mathit{x}\mathit{y}\mathbf{+}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{\times }\sqrt{\mathbf{1}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}}$

$={\cos }^{-1}\left[\frac{7}{25}\times \frac{24}{25}+{\sqrt{1-\left(\frac{7}{25}\right)}}^2\times {\sqrt{1-\left(\frac{24}{25}\right)}}^2\right]$

$={\cos }^{-1}\left[\frac{7}{25}\times \frac{24}{25}–\sqrt{1–\left(\frac{49}{625}\right)\times }\sqrt{[1–\left(\frac{576}{625}\right)}\right]$

$={\cos }^{-1}\left[\frac{168}{625}–\sqrt{\left(\frac{576}{625}\right)\times }\sqrt{\left(\frac{49}{625}\right)}\right]$

$={\cos }^{-1}\left[\frac{168}{625}–\left(\frac{24}{25}\right)\times \left(\frac{7}{25}\right)\right]$

$={\cos }^{-1}\left(0\right)$

$$\begin{gathered} ={\cos }^{-1}\left(\cos \frac{\mathrm{\pi }}{2}\right) \\ =\frac{\mathbf{\pi }}{\mathbf{2}} \end{gathered}$$ $\left[\mathbf{\because }\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\left[x\right]\mathbf{}\mathbf{\in }\mathbf{}\mathbf{[}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{1}\mathbf{]}\right]$

Hence, Option ‘A’ is Correct.