Question #2c948

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Answer 1 · 2017-02-26T23:57:58

See below.

Easiest way, Euler's Formula:

$e^{i θ} = cos θ + i sin θ$ $e^(i theta) = cos theta + i sin theta$

${(e^{i θ})}^{2} = e^{i 2 θ}$ $(e^(i theta))^2 = e^(i 2 theta)$

$= cos 2 theta + i sin 2 theta qquad triangle$

$= (cos theta + i sin theta)^2$

Expanding that out....
$= cos^2 theta - sin^2 theta + i \ 2 sin theta cos theta qquad star$

Comparing real and imaginary terms in $star$ and $triangle$ :

$cos 2 theta = cos^2 theta - sin^2 theta$

$color(blue)(i) sin 2 theta = color(blue)(i) 2 sin theta cos theta$

Ta-da!!