What is the simplified value of `(2sin^(3)theta-sintheta)/(costheta-2cos^(3)theta)`?

Options

A. `tantheta`
B. `sintheta`
C. `costheta`
D. `cottheta`

Answer

To simplify the expression \(\frac{2\sin^3\theta - \sin\theta}{\cos\theta - 2\cos^3\theta}\), we can follow these steps:

### Step 1: Factor out common terms in the numerator and denominator
In the numerator, we can factor out \(\sin\theta\):
\[ 2\sin^3\theta - \sin\theta = \sin\theta(2\sin^2\theta - 1) \]
In the denominator, we can factor out \(\cos\theta\):
\[ \cos\theta - 2\cos^3\theta = \cos\theta(1 - 2\cos^2\theta) \]
So, we can rewrite the expression as:
\[ \frac{\sin\theta(2\sin^2\theta - 1)}{\cos\theta(1 - 2\cos^2\theta)} \]

### Step 2: Use trigonometric identities
Recall the identity \(2\sin^2\theta - 1 = \cos(2\theta)\) and \(1 - 2\cos^2\theta = -\cos(2\theta)\). Thus, we can rewrite the expression:
\[ \frac{\sin\theta \cos(2\theta)}{\cos\theta (-\cos(2\theta))} \]

### Step 3: Simplify the expression
Now, we can simplify the expression:
\[ \frac{\sin\theta \cos(2\theta)}{\cos\theta (-\cos(2\theta))} = -\frac{\sin\theta}{\cos\theta} \]
This simplifies to:
\[ -\tan\theta \]

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