# What does $2sin^{-1} \frac{3}{5}$ reduce to?

Toolbox:
• $$sin^{-1}x=tan^{-1}\frac{x}{\sqrt{1-x^2}}$$
• $$2tan^{-1}x=tan^{-1}\frac{2x}{1-x^2} |x| < 1$$
Given $2sin^{-1} \frac{3}{5}$, Let $x =$ $$\frac{3}{5}$$, $$\rightarrow \large \frac{x}{\sqrt{1-x^2}}=\frac{\frac{3}{5}}{\sqrt{1-\frac{9}{25}}}=\frac{3}{4}$$
$$\Rightarrow\:sin^{-1}\frac{3}{5}=tan^{-1}\frac{3}{4}$$
$$\Rightarrow 2sin^{-1}\frac{3}{5}=2tan^{-1}\frac{3}{4}$$
Given $$2tan^{-1}x=tan^{-1}\frac{2x}{1-x^2}$$, we need to evaluate $$2tan^{-1}\frac{3}{4}$$
Let $x =\frac{3}{4} \rightarrow \large \frac{2x}{1-x^2}=\frac{2.\frac{3}{4}}{1-\frac{9}{16}}=\frac{6}{4}.\frac{16}{7}=\frac{24}{7}$
$$\Rightarrow\:2tan^{-1}\frac{3}{4} = tan^{-1}\frac{24}{7}$$