Mathematics

Problem

Consider real numbers

x

and

y

such that

x,\,y\in[-1,1]

and

x^{2}+y^{2}\le 1

. a. (i) Prove that

\arcsin x+\arcsin y=\arcsin\bigl(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\bigr).

Clearly mention any domain restrictions required for the identity to hold in the principal value branch. b. (ii) Show that when

x=\dfrac{5}{13}

and

y=\dfrac{12}{13}

, the right‐hand side of the identity in part (i) simplifies to

\dfrac{\pi}{2}

. c. (iii) Using the identity, evaluate

\arcsin\left(\dfrac{5}{13}\right)+\arcsin\left(\dfrac{12}{13}\right)

without a calculator.