If `1+4sqrt(3)i=(a+ib)^2`, prove that `a^2-b^2=1 and ab = 2sqrt(3)`

If 1 +4 sqrt3i = (a+ ib)^2 , prove that : a^2-b^2=1 and ab=2 sqrt3 .

If x+iy=(a+ib)/(a-ib) , then prove that x^2+y^2=1

If a+ib= (c+i)/(c-i) , where c is real, prove that a^2+b^2=1 and b/a= (2c)/(c^2-1) .

If x-iy= sqrt((a-ib)/(c-id)) , prove that (x^2+y^2)^2= (a^2+b^2)/(c^2+d^2) .

If y = sqrt(1+sqrt(1+x^4)) , prove that y(y^2-1)dy/dx = x^3

If x-iy= sqrt((a-ib)/(c-id)) prove that (x^2+y^2)^2= (a^2+b^2)/(c^2+d^2) .

Prove that : cosec[tan^-1(-sqrt3)] = -2/sqrt3

If sqrt(1-x^6)+sqrt(1-y^6)=a^3(x^3-y^3) , prove that : dy/dx = x^2/y^2sqrt((1-y^6)/(1-x^6)) .

Prove that sin^-1 (-(sqrt3)/(2)) = - (pi/3)

If sqrt(1 - x^6) + sqrt(1 - y^6) = a^3 (x^3 - y^3) prove that (dy)/(dx) = (x^2 sqrt(1 - y^6))/(y^2 sqrt(1 - x^6))