Prove the following identity, where the angles involved are acute angles for which the expressions are defined.
(vii) `(sintheta-2sin^3theta)/(2cos^3theta-costheta)=tantheta`

Explanation

To prove the identity \(\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta\), we will start with the left-hand side (LHS) and simplify it step by step.

### Step 1: Write down the LHS
\[ \text{LHS} = \frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} \]

### Step 2: Factor out common terms
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