Question

The value of ${\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}$ is:

• A. ${\cos }^{-1}\frac{60}{85}$
• B. ${\cos }^{-1}\frac{24}{85}$
• C. ${\cos }^{-1}\frac{84}{85}$
• D. ${\cos }^{-1}\frac{15}{17}$

Answer 1

The correct option is C ${\cos }^{-1}\frac{84}{85}$

We have,
${\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}$
$={\cos }^{-1}\frac{4}{5}-{\cos }^{-1}\frac{15}{17}$
$\{\because {\sin }^{-1}\frac{3}{5}={\cos }^{-1}\frac{4}{5},{\sin }^{-1}\frac{8}{17}={\cos }^{-1}\frac{15}{17}\}$
Using the formula:
$\{{\cos }^{-1}x-{\cos }^{-1}y={\cos }^{-1}\{xy+\sqrt{1-x^2}\sqrt{1-y^2}\},x\ge 0,y\ge 0,x\le y\}$
$={\cos }^{-1}\{\frac{4}{5}\times \frac{15}{17}+\sqrt{1-{\left(\frac{4}{5}\right)}^2}\times \sqrt{1-{\left(\frac{15}{17}\right)}^2}\}$
$={\cos }^{-1}\{\frac{4}{5}\times \frac{15}{17}+\frac{3}{5}\times \frac{8}{17}\}$
$={\cos }^{-1}\{\frac{60}{85}+\frac{24}{85}\}$
$={\cos }^{-1}\frac{84}{85}$