Final answer:
To verify the trigonometric identity, we would use various trigonometric formulas to transform the sums of sines and cosines into a form that matches cot2x.
Step-by-step explanation:
To verify the identity cos3x - cos2x + cosx / sin3x - sin2x + sinx = cot2x, we can employ various trigonometric identities to simplify the given expression and show that it's equivalent to cot2x. One such identity is sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) - sin²(θ).
Furthermore, using the sum-to-product trigonometric identities: sin α + sin β = 2 sin((α + β)/2) cos((α - β)/2) and cos α + cos β = 2 cos((α + β)/2) cos((α - β)/2), we can transform the sums of sines and cosines in the numerator and denominator into products that can simplify further.
By doing this, we should eventually show that the original expression does indeed simplify to cot2x, thus verifying the identity.