(a) Differentiate sec^-1(1/(√(1- x^2))) w.r.t. sin^-1(2 x √(1- x^2)).

Let $\displaystyle \mathrm{x}=\sin \theta$. Then $$\begin{aligned} & \mathrm{y}=\tan \mathrm{x}+\sec \mathrm{x}=\frac{\sin \mathrm{x}+1}{\cos \mathrm{x}} \\ & \Rightarrow \frac{\mathrm{d} \mathrm{y}}{\mathrm{d} \mathrm{x}}=\frac{\cos \mathrm{x}(\cos \mathrm{x})+(\sin \mathrm{x}+1) \sin \mathrm{x}}{\cos ^{2} \mathrm{x}} \\ & \quad=\frac{\cos ^{2} \mathrm{x}+\sin ^{2} \mathrm{x}+\sin \mathrm{x}}{\cos ^{2} \mathrm{x}}=\frac{1+\sin \mathrm{x}}{1-\sin ^{2} \mathrm{x}}=\frac{1}{1-\sin \mathrm{x}} \\ & \Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}=\frac{(1-\sin \mathrm{x}) \cdot 0-1(0-\cos \mathrm{x})}{(1-\sin \mathrm{x})^{2}}=\frac{\cos \mathrm{x}}{(1-\sin \mathrm{x})^{2}} \end{aligned} $$