Prove the trigonometric identitycos(3θ) = 4 cos3(θ) − 3 cos(θ)

Sure, here is the step-by-step proof of the trigonometric identity cos(3θ) = 4 cos³(θ) − 3 cos(θ):

Step 1: We start with the formula for cos(3θ), which is cos(3θ) = cos(2θ + θ).

Step 2: We can then use the addition formula for cosine, which is cos(A + B) = cos(A)cos(B) - sin(A)sin(B). So, cos(2θ + θ) = cos(2θ)cos(θ) - sin(2θ)sin(θ).

Step 3: We know that cos(2θ) = 2cos²(θ) - 1 and sin(2θ) = 2sin(θ)cos(θ). Substituting these into the equation from step 2, we get cos(3θ) = (2cos²(θ) - 1)cos(θ) - 2sin(θ)cos(θ)sin(θ).

Step 4: We also know that sin²(θ) = 1 - cos²(θ). Substituting this into the equation from step 3, we get cos(3θ) = 2cos³(θ) - cos(θ) - 2cos(θ)(1 - cos²(θ)).

Step 5: Simplifying the equation from step 4, we get cos(3θ) = 2cos³(θ) - cos(θ) - 2cos(θ) + 2cos³(θ).

Step 6: Combining like terms, we get cos(3θ) = 4cos³(θ) - 3cos(θ), which is the desired identity.