SOLUTION: Prove that {{{ (-1+i*sqrt(3))^n + (-1-i*sqrt(3))^n }}} has either the value {{{ 2^(n+1) }}} or the value {{{ - 2^n }}} if n is any integer (positive, negative or zero).

SOLUTION: Prove that {{{ (-1+isqrt(3))^n + (-1-isqrt(3))^n }}} has either the value {{{ 2^(n+1) }}} or the value {{{ - 2^n }}} if n is any integer (positive, negative or zero).

Algebra.Com

Question 1161849: Prove that

has either the value or the value if n is any integer (positive, negative or zero).
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Because cosine is an even function and sine is an odd function,

Then

is equal to

The terms in sine cancel, leaving

If n is a multiple of 3, then , , and the value of the expression is

If the integer n is not a multiple of 3, then or , , and the value of the expression is

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