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

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

(-i)^(4n+3), where n Is a positive integer.

((1+i)/(1-i))^(4n+1) where n is a positive integer.

Solve (x-1)^(n)=x^(n), where n is a positive integer.

If n is be a positive integer,then (1+i)^(n)+(1-i)^(n)=2^(k)cos((n pi)/(4)), where k is equal to

Show that (-sqrt(-1))^(4n+3) =i , where n is a positive integer.

If n be a positive integer, then prove that (1+i)^n+(1-i)^n=2^(n/2+1)."cos"((npi)/4)

Prove that 2^(n)>1+n sqrt(2^(n-1)),AA n>2 where n is a positive integer.

What are the limits of (2^(n)(n+1)^(n))/(n^(n)) , where n is a positive integer ?