30. Prove i) \( 2 \sin ^{-1} x=\sin ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\},-\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}} \). [M-2015, S-2020, ii) \( 2 \cos ^{-1} x=\sin ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}, \frac{1}{\sqrt{2}} \leq x \leq 1 \). [M-2016, M-2017] 31. Prove i) \( 3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right), x \in\left[-\frac{1}{2}, \frac{1}{2}\right] \cdot[J-2016] \) ii) \( 3 \cos ^{-1} x=\sin ^{-1}\left(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right][ M -2014(3 M)] \)

Question

30. Prove i) \( 2 \sin ^{-1} x=\sin ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\},-\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}} \). [M-2015, S-2020, ii) \( 2 \cos ^{-1} x=\sin ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}, \frac{1}{\sqrt{2}} \leq x \leq 1 \). [M-2016, M-2017] 31. Prove i) \( 3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right),eft(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right][ M -2014(3 M)] \)