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

If x and y are positive real numbers and m, n are any positive integers, then Prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt =1/4

If n is a positive integer and r is a nonnegative integer, prove that the coefficients of x^r and x^(n-r) in the expansion of (1+x)^(n) are equal.

If n_1, n_2 are positive integers, then (1 + i)^(n_1) + ( 1 + i^3)^(n_1) + (1 + i_5)^(n_2) + (1 + i^7)^(n_2) is real if and only if :

If a and b are distinct Integers, prove that a - b is a factor of a^(n) - b^(n) , whenever n is a positive integer. [Hint: write a^(n) = (a-b + b)^(n) and expand]

Prove that (a) (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer. (b) (1+isqrt(3))^n+(1-isqrt(3)^n=2^(n+1)cos((npi)/3) , where n is a positive integer

If z_1a n dz_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

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

If a and b are distinct integers, prove that a-b is a factor of a^n - b^n , whenever n is a positive integer.

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_m-2n(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))