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Addition formula for the sine
An addition in the argument of the sine gives
Explanation
From the definition of the sine follows
Substitution with Euler's formula gives
so that
Example 1
For sin (2x) applies
sin (2x) = sin (x + x) = sin x cos x + cos x sin x = 2 sin x cos x
You get the same result with the formula for the sine of multiples, of course.