Let’s first deal with double-angle formulas, which break an expression involving a single angle into one involving two angles, and prove at least one of them. The proof is short and easy, which is a good starting point for this chapter.

Prove that sin2*u* = 2sin*u*cos*u*.

To prove this identity, let *v* = *u* in the sum formulas for sines. Then obtain the following:

sin2*u* = sin(*u* + *u*) = sin*u*cos*u* + cos*u*sin*u* = 2sin*u*cos*u*

Note that the order of the factors does not change the product; thus sin*u*cos*u* = cos*u*sin*u*.

You proved your first double-angle formula. The other double-angle formulas can be proved in a similar way. Let’s list all three of them.

The following are called double-angle formulas:

sin2*u* = 2sinucos*u*

cos2*u* = cos^{2}*u* – sin^{2}*u* = ...

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