Question

$si{n}^2\left(co{s}^{-1}\frac{1}{2}\right)+co{s}^2\left(si{n}^{-1}\frac{1}{3}\right)$ is equal to

• A. $\frac{13}{36}$
• B. $\frac{59}{36}$
• C. $\frac{5}{36}$
• D. none of these

Answer 1

The correct option is B $\frac{59}{36}$

$si{n}^2\left(co{s}^{-1}\frac{1}{2}\right)+co{s}^2\left(si{n}^{-1}\frac{1}{3}\right)$

$co{s}^{-1}\frac{1}{2}=si{n}^{-1}\sqrt{1-\frac{1}{4}}=si{n}^{-1}\frac{\sqrt{3}}{2}$

$si{n}^{-1}\frac{1}{3}=co{s}^{-1}\sqrt{1-\frac{1}{9}}=co{s}^{-1}\frac{2\sqrt{2}}{3}$

$=si{n}^2\left(si{n}^{-1}\frac{\sqrt{3}}{2}\right)+co{s}^2\left(co{s}^{-1}\frac{2\sqrt{2}}{3}\right)$

$={\left(\frac{\sqrt{3}}{2}\right)}^2+{\left(\frac{2\sqrt{2}}{3}\right)}^2=\frac{3}{4}+\frac{8}{9}$

$=\frac{27+32}{36}=\frac{59}{36}$