Simplify sin4theta

Answer 1

Using trigonometric identities, we can simplify sin(4θ) as follows:

sin(4θ) = 2sin(2θ)cos(2θ)

Since sin(2θ) = 2sin(θ)cos(θ), we can substitute this identity into the expression:

sin(4θ) = 2(2sin(θ)cos(θ))(cos(2θ))

Expanding further using the double angle formula for cosine:

cos(2θ) = cos^2(θ) - sin^2(θ)

sin(4θ) = 2(2sin(θ)cos(θ))(cos^2(θ) - sin^2(θ))

Using the identity sin^2(θ) + cos^2(θ) = 1:

sin(4θ) = 2(2sin(θ)cos(θ))(1 - sin^2(θ))

Expanding further:

sin(4θ) = 4sin(θ)cos(θ) - 4sin^3(θ)cos(θ)

Therefore, sin(4θ) = 4sin(θ)cos(θ) - 4sin^3(θ)cos(θ).