Prove the double angle formula for sine.

The double angle formula for sine is sin(2θ) = 2sin(θ)cos(θ).

To prove the double angle formula for sine, we start with the sum formula for sine:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Let A = B = θ, then we have:

sin(2θ) = sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ)

Using the commutative property of multiplication, we can rewrite this as:

sin(2θ) = 2sin(θ)cos(θ)

Therefore, we have proven the double angle formula for sine.