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

Options

A. `tan theta`
B. `sin theta`
C. `cos theta`
D. `cot theta`

Answer

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

### Step 1: Factor the numerator
The numerator \(2 \sin^3 \theta - \sin \theta\) can be factored:
\[ \sin \theta (2 \sin^2 \theta - 1) \]
This is because we can take out \(\sin \theta\) as a common factor.

### Step 2: Factor the denominator
The denominator \(\cos \theta - 2 \cos^3 \theta\) can also be factored:
\[ \cos \theta (1 - 2 \cos^2 \theta) \]
This is because we can take out \(\cos \theta\) as a common factor.

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

### Step 4: Use trigonometric identities
We know that:
\[ 2 \sin^2 \theta - 1 = \cos 2\theta \]
and
\[ 1 - 2 \cos^2 \theta = -\cos 2\theta \]
Thus, we can substitute these identities into our expression:
\[ \frac{\sin \theta \cos 2\theta}{\cos \theta (-\cos 2\theta)} \]

### Step 5: Simplify the expression
The \(\cos 2\theta\) terms cancel out (assuming \(\cos 2\theta \neq 0\)):
\[ \frac{\sin \theta}{-\cos \theta} = -\tan \theta \]

### Final Result
Thus, the simplified value of the expression is:
\[ -\tan \theta \]