The value of `(sin theta - 2 sin^3 theta)/(2 cos^3 theta-cos theta)` is

Answer

To solve the expression \(\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta}\), we will simplify it step by step.

### Step 1: Factor the numerator
The numerator can be factored by taking \(\sin \theta\) common:
\[ \sin \theta - 2 \sin^3 \theta = \sin \theta (1 - 2 \sin^2 \theta) \]

### Step 2: Factor the denominator
The denominator can be factored as well:
\[ 2 \cos^3 \theta - \cos \theta = \cos \theta (2 \cos^2 \theta - 1) \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{\sin \theta (1 - 2 \sin^2 \theta)}{\cos \theta (2 \cos^2 \theta - 1)} \]

### Step 4: Use trigonometric identities
Recall the double angle identities:
- \(2 \sin^2 \theta = 1 - \cos 2\theta\)
- \(2 \cos^2 \theta = 1 + \cos 2\theta\)

Using these identities, we can rewrite \(1 - 2 \sin^2 \theta\) and \(2 \cos^2 \theta - 1\):
\[ 1 - 2 \sin^2 \theta = \cos 2\theta \]
\[ 2 \cos^2 \theta - 1 = \cos 2\theta \]

### Step 5: Substitute back into the expression
Substituting these identities back into our expression gives:
\[ \frac{\sin \theta \cos 2\theta}{\cos \theta \cos 2\theta} \]

### Step 6: Cancel common terms
Since \(\cos 2\theta\) appears in both the numerator and the denominator, we can cancel it (assuming \(\cos 2\theta \neq 0\)):
\[ \frac{\sin \theta}{\cos \theta} = \tan \theta \]

### Final Answer
Thus, the value of the expression \(\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta}\) is:
\[ \tan \theta \]

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