Question #2c948

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Answer 1 · 2017-02-26T23:57:58

See below.

Easiest way, Euler's Formula:

#e^(i theta) = cos theta + i sin theta#

#(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!!