if `(1+i x/a) (1+i x/b)(1+i x/c)…=A+iB` then prove that `(1+ x^2/a^2)(1+x^2/b^2)(1+x^2/c^2)…=A^2+B^2 `

If ((x+i)^(2))/(3x+2)=a+ib, then prove that ((x^(2)+1)^(2))/((3x+2)^(2))=a^(2)+b^(2)

If x =1/2 (sqrt(a/b)-sqrt(b/a)) then prove that (2asqrt(1+x^2))/(x+sqrt(1+x^2))= a +b

If (a_1+ib_1)(a_2+ib_2)...(a_n+ib_n) = x+iy , prove that : (a_1^2+b_1^2)(a_2^2+b_2^2)...(a_n^2+b_n^2)=x^2+y^2 .

If x+iy=(1+2i)/(2+i) prove that x^(2)+y^(2)=1

If a+ib=((x+i)^(2))/(2x^(2)+1), prove that a^(2)+b^(2)=((x^(2)+1)^(2))/((2x^(2)+1)^(2))

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 a+ib=((x+i)^(2))/(2x^(2)+1) prove that a^(2)+b^(2)=((x^(2)+1)^(2))/((2x^(2)+1)^(2))

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

If the sum of the roots of the equation (1)/(x+a)+(1)/(x+b)=1/c is zero,the prove that the product of the root is (-(1)/(2))(a^(2)+b^(2))

If a^(2) + b^(2) + c^(2) = -2 and f(x) = |(1 + a^(2)x,(1 + b^(2))x,(1 + c^(2)) x),((1 + a^(2))x,1 + b^(2)x,(1 + c^(2))x),((1 + a^(2)) x,(1 + b^(2))x,1 + c^(2)x)| , then f(x) is a polynomial of degree