$\"\"$

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Prove that DeMoivres theorem $\"\"$ .

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Consider the statment is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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$\"\"$ is true for $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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$\"\"$ is true for positive integer $\"\"$ .

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$\"\"$

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Show that $\"\"$ must be true.

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Apply the property : $\"\"$ .

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Apply the property : $\"\"$ .

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$\"\"$

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$\"\"$

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The final statement is exactly $\"\"$ , so $\"\"$ is true.

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Because $\"\"$ is true for $\"\"$ and $\"\"$ implies $\"\"$ , $\"\"$ is true for $\"\"$ and so on.

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That is, by the principle of mathematical induction, $\"\"$ is true for all positive integers $\"\"$ .

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$\"\"$

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$\"\"$ is true for all positive integers $\"\"$ .