If `(x+i y)^5=p+i q ,` then prove that `(y+i x)^5=q+i pdot`

If (x+i y)(p+i q)=(x^2+y^2)i , prove that x=q ,y=pdot

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

If x= cos alpha+ i sin alpha, y = cos beta+ i sin beta, then prove that (x-y)/(x+y) =i (tan )(alpha-beta)/2

If x+i y=(a+i b)/(a-i b) prove that x^2+y^2=1

If X=[[0, 1], [0, 0]] , then prove that (p I+q X)^m=p^m I+m p^(m-1)q X ,AAp ,q in R ,where I is a two rowed unit matrix and m in N.

If the normals at four points P (x_i y_i), i = 1, 2, 3, 4 on the rectangular hyperbola xy = c^2 , meet at the point Q(h, k), then prove that x_1 + x_2 + x_3 + x_4 =h

If x^(p) y^(q) = (x + y)^((p + q)) " then " (dy)/(dx)= ?

If I is the set of real numbers, then prove that the relation R={(x,y) : x,y in I and x-y " is an integer "} , then prove that R is an equivalence relations.

If sqrt(a+i b)=x+i y ,\ then possible value of sqrt(a-i b) is a. x^2+y^2 b. sqrt(x^2+y^2) c. x+i y d. x-i y e. sqrt(x^2-y^2)

If the normal at four points P_(i)(x_(i), (y_(i)) l, I = 1, 2, 3, 4 on the rectangular hyperbola xy = c^(2) meet at the point Q(h, k), prove that x_(1) + x_(2) + x_(3) + x_(4) = h, y_(1) + y_(2) + y_(3) + y_(4) = k x_(1)x_(2)x_(3)x_(4) =y_(1)y_(2)y_(3)y_(4) =-c^(4)