Prove that

\\sin^{-1}x+sin^{-1}y=\pi-sin^{-1}z \\sin(sin^{-1}x+sin^{-1}y)=sin(\pi-sin^{-1}z) \\x \cos(sin^{-1}y)+y \cos(sin^{-1}x) = z \\x \cos(sin^{-1}y)+y \cos(sin^{-1}x) = z \\x \sqrt{1-y^2}+y \sqrt{1-x^2} = z

Square both sides.. and complete the proof by squaring agian

\\x^2(1-y^2)+y^2(1-x^2)+2xy\sqrt{(1-x^2)(1-y^2)}=z^2 \\2xy\sqrt{(1-x^2)(1-y^2)}=z^2-x^2-y^2+2x^2y^2 \\2x^2y^2((1-x^2)(1-y^2))=z^4+x^4+y^4+4x^4y^4+2x^2y^2-2x^2z^2-2z^2y^2+.... \text{I am lazy to type more) but youwill get the result :) }