The given identity is: sin(2θ) = 2sin(θ)cos(θ) / 1 - cos^2(θ)



To prove this identity, we will use the double angle formula for sine:

sin(2θ) = 2sin(θ)cos(θ)

We will also use the Pythagorean identity:

sin^2(θ) + cos^2(θ) = 1



Start with the expression sin(2θ).



Apply the double angle formula for sine:

sin(2θ) = 2sin(θ)cos(θ)



Multiply the numerator and denominator by 1 + cos(2θ):

sin(2θ) = (2sin(θ)cos(θ))(1 + cos(2θ)) / (1 + cos(2θ))



Expand the denominator using the double angle formula for cosine:

sin(2θ) = (2sin(θ)cos(θ))(1 + cos^2(θ) - sin^2(θ)) / (1 + 2cos^2(θ) - 1)

Simplify the denominator:

sin(2θ) = (2sin(θ)cos(θ))(cos^2(θ) - sin^2(θ)) / (2cos^2(θ))



Rearrange the terms in the numerator:

sin(2θ) = 2sin(θ)cos(θ)(-sin^2(θ) + cos^2(θ)) / (2cos^2(θ))



Combine like terms in the numerator:

sin(2θ) = 2sin(θ)cos(θ)(cos^2(θ) - sin^2(θ)) / (2cos^2(θ))



Use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to simplify the numerator:

sin(2θ) = 2sin(θ)cos(θ)(1 - sin^2(θ)) / (2cos^2(θ))



Simplify further:

sin(2θ) = 2sin(θ)cos(θ)(1 - sin^2(θ)) / (2cos^2(θ))



Cancel out the common factors in the numerator and denominator:

sin(2θ) = sin(θ)(1 - sin^2(θ)) / cos^2(θ)



Replace sin^2(θ) with 1 - cos^2(θ) using the Pythagorean identity:

sin(2θ) = sin(θ)(1 - (1 - cos^2(θ))) / cos^2(θ)

Simplify the expression:

sin(2