If `z=(costheta+isintheta)`, show that `z^(n)+(1)//(z^(n))=2cosntheta and `z^(n)-(1)(z^(n))=2isinntheta

If z+1//z=2costheta, prove that |(z^(2n)-1)//(z^(2n)+1)|=|tanntheta|

Prove that (1+costheta+isintheta)^(n)+(1+costheta-isintheta)^(n)=2^(n+1)cos^(n)((theta)/(2))cos((ntheta)/(2))

choose the correct pair of statement : (i) (costheta+isintheta)^(n)=cosntheta+isinntheta , x is an integer (ii) (sintheta+icostheta)^(n)=sinntheta+icosthetantheta , n is an integer (iii) If z=costheta+isintheta , then z^(-1)=costheta-isintheta (iv)(z+barz)/(2)=Re(z)

If (costheta +isintheta)(cos2theta +isin2theta).....(cosntheta + isinntheta) = 1 then the value of theta is:

Using the mathematical induction, show that for any natural number n, with the assumption i^(2)=-1 , (r(costheta+isintheta))^(n)=r^(n)(cosntheta+isinntheta)

If n is a positive integer, prove that |I m(z^n)|lt=n|Im(z)||z|^(n-1.)

If n >1 , show that the roots of the equation z^n=(z+1)^n are collinear.

Prove by mathematical induction (costheta+isintheta)^(n)=cosntheta+isinntheta where i=sqrt(-1) .

zo is one of the roots of the equation z^n cos theta0+ z^(n-1) cos theta2 +. . . . . . + z cos theta(n-1) + cos theta(n) = 2, where theta in R , then (A) |z0| 1/2 (C) |z0| = 1/2