If `"sin"^(-1)(x/a) + "sin"^(-1) (y/b)=alpha "prove that" x^2/a^2+(2xy)/(ab) "cos" alpha+y^2/b^2 ="sin"^2 alpha`

If "cos"^(-1)(x/a) ="cos"^-1(y/b) = theta, "prove that" x^2/a^2 -(2xy)/(ab) "cos" theta +y^2/b^2 ="sin"^2 theta.

If "cos"^(-1)(x/y) +"cos"^-1(y/3)= theta, "prove that" 9x^2- 12xy "cos" theta +4y^2 =36 "sin"^2 theta .

If "sin"^(-1)x + "sin"^(-1)y +"sin"^(-1)z = pi "prove that" x^4+y^4+z^4 +4x^2y^2z^2=2(x^2y^2+y^2z^2+z^2x^2)

If the straight line x cos alpha + y sin alpha = p touches the curve (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 , then prove that a^(2) cos^(2) alpha + b^(2) sin^(2) alpha = p^(2) .

If sin^(-1)((x)/(a)) +sin^(-1)((y)/(b)) = sin^(-1)((c^(2))/(ab)) , then prove that b^(2)x^(2)+2xy sqrt(a^(2)b^(2)-c^(4)) +a^(2)y^(2)=c^(4)

If y =(sin^(-1) x)^2 , prove that (1-x^2)y_2 - xy_1 - 2 = 0