Proof:

We are given the equation: (sinA - 2sin³A) = (2cos³A - cosA)tanA.

To prove this, we can simplify both sides of the equation separately and show that they are equal.

Simplifying the left side:
(sinA - 2sin³A)

We can first factor out sinA from both terms:
sinA(1 - 2sin²A)

Using the identity sin²A + cos²A = 1, we can substitute cos²A = 1 - sin²A into the expression:
sinA(1 - 2(1 - cos²A))
sinA(1 - 2 + 2cos²A)
sinA(2cos²A - 1)

Simplifying the right side:
(2cos³A - cosA)tanA

Using the identity cos²A = 1 - sin²A, we can rewrite cos³A as (1 - sin²A)cosA:
(2(1 - sin²A)cosA - cosA)tanA
(2cosA - 2sin²AcosA - cosA)tanA
(2cosA(1 - sin²A) - cosA)tanA
(2cosAcos²A - cosA)tanA

Using the identity cos²A = 1 - sin²A, we can simplify further:
(2(1 - sin²A)cosA - cosA)tanA
(2cos²A - cosA)tanA

Comparing the left and right sides:
We have simplified the left side to sinA(2cos²A - 1) and the right side to (2cos²A - cosA)tanA.

Since both sides now have the same expression of (2cos²A - 1), we can conclude that the given equation is true:

(sinA - 2sin³A) = (2cos³A - cosA)tanA.

Therefore, we have successfully proven the given equation.