Proof:

We are given the expression:

tanA = sinA - 2sin3A / 2cos3A - cosA

To prove this expression, we'll start by simplifying the right-hand side (RHS) of the equation.

Simplifying the RHS:

We can rewrite the expression as follows:

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

Next, we'll simplify the numerator and denominator separately.

Simplifying the numerator:

Using the trigonometric identity sin3A = 3sinA - 4sin^3A, we can rewrite the numerator as:

sinA - 2sin3A = sinA - 2(3sinA - 4sin^3A)
= sinA - 6sinA + 8sin^3A
= -5sinA + 8sin^3A

Simplifying the denominator:

Using the trigonometric identity cos3A = 4cos^3A - 3cosA, we can rewrite the denominator as:

2cos3A - cosA = 2(4cos^3A - 3cosA) - cosA
= 8cos^3A - 7cosA

Substituting the simplified numerator and denominator back into the expression:

Now, we can substitute these simplified forms back into the original expression:

tanA = (-5sinA + 8sin^3A) / (8cos^3A - 7cosA)

To further simplify this expression, we'll use the trigonometric identity tanA = sinA / cosA:

tanA = (sinA / cosA) * ((-5sinA + 8sin^3A) / (8cos^3A - 7cosA))

Now, we can cancel out the common factors:

tanA = (-5sinA + 8sin^3A) / (8cos^3A - 7cosA)

Therefore, we have proven that:

tanA = sinA - 2sin3A / 2cos3A - cosA