# Difference between revisions of "1981 AHSME Problems/Problem 24"

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<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)</cmath> | <cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)</cmath> | ||

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− | + | <math>=\boxed{\textbf{2\cos(n\theta)}},</math> | |

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+ | which gives the answer <math>\boxed{\textbf{D}}.</math> |

## Latest revision as of 15:24, 6 July 2021

## Problem

If is a constant such that and , then for each positive integer , equals

## Solution

Multiply both sides by and rearrange to . Using the quadratic equation, we can solve for . After some simplifying:

Substituting this expression in to the desired gives:

Using DeMoivre's Theorem:

Because is even and is odd:

$=\boxed{\textbf{2\cos(n\theta)}},$ (Error compiling LaTeX. ! Missing $ inserted.)

which gives the answer