How do you simplify sin²θ to non-exponential trigonometric functions?

You can simplify the expression sin(2θ) to non-exponential trigonometric functions using trigonometric identities. One of the most commonly used identities for this purpose is the double-angle identity for sine:

sin(2θ) = 2sin(θ)cos(θ).

Here's how you can derive this identity:

Start with the sum-to-product identity for sine:

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

Set A = B = θ:

sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ).

Simplify the left side:

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

Notice that sin(2θ) is equal to the right side of the equation:

sin(2θ) = 2sin(θ)cos(θ).

So, sin(2θ) can be simplified to 2sin(θ)cos(θ) using the double-angle identity for sine. This expression involves only non-exponential trigonometric functions, namely sin(θ) and cos(θ).