Show that the roots of equation `(1+z)^n=(1-z)^n are -i tan (rpi/n), r=0,1,2,,…………,(n-1)` excluding the vlaue when n is even and `r=n/2`

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

If the roots of (z-1)^n=i(z+1)^n are plotted in ten Arg and plane, then prove that they are collinear.

If z_(1),z_(2),z_(3),…,z_(n-1) are the roots of the equation z^(n-1)+z^(n-2)+z^(n-3)+…+z+1=0 , where n in N, n gt 2 and omega is the cube root of unity, then

Evaluate sum_(r=1)^(n-1) cos^(2) ((rpi)/(n)) .

If z_1,z_2,z_3,………..z_(n-1) are the roots of the equation 1+z+z^2+…….+z^(n-1)=0, where n epsilon N, ngt2 then (A) z_1,z_2, …z_(n-1) are terms of a G.P. (B) z_1,z_2,……,z_(n-1) are terms of an A.P. (C) |z_1|=|z_2|=|z_3|=.|z_(n-1)|!=1 (D) none of these

Show that: sum_(r=0)^(n-1)|z_1+alpha^r z_2|^2=n(|z_1|^2+|z_2|^2),w h e r e, alpha ; r=0,1,2,...,(n-1) , are the nth roots of unity and z_1, z_2 are any two complex numbers.

If roots of an equation x^n-1=0a r e1,a_1,a_2,..... a_(n-1), then the value of (1-a_1)(1-a_2)(1-a_3)(1-a_(n-1)) will be n b. n^2 c. n^n d. 0

If n be integer gt1, then prove that sum_(r=1)^(n-1) cos (2rpi)/n=-1

If n in N >1 , then the sum of real part of roots of z^n=(z+1)^n is equal to

zo is one of the roots of the equation z^n cos theta_0+ z^(n-1) cos theta_2 +. . . . . . + z cos theta_(n-1) + cos theta_(n) = 2 , where theta in R , then (A) |z_0| lt 1/2 (B) |z_0| gt 1/2 (C) |z_0| = 1/2 (D)None of these