What is the value of
`((sinA-2sin^(3)A))/((2cos^(3)A-cosA))`?

Options

A. cot A
B. cosec A
C. sec A
D. tan A

Answer

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

### Step 1: Factor the numerator and the denominator
The numerator can be factored as follows:
\[ \sin A - 2\sin^3 A = \sin A(1 - 2\sin^2 A) \]
The denominator can be factored as:
\[ 2\cos^3 A - \cos A = \cos A(2\cos^2 A - 1) \]

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

### Step 3: Use trigonometric identities
We know from trigonometric identities that:
\[ 1 - 2\sin^2 A = \cos 2A \quad \text{and} \quad 2\cos^2 A - 1 = \cos 2A \]
Thus, we can substitute these identities into our expression:
\[ \frac{\sin A \cos 2A}{\cos A \cos 2A} \]

### Step 4: Simplify the expression
Since \(\cos 2A\) is common in both the numerator and the denominator, we can cancel it out (assuming \(\cos 2A \neq 0\)):
\[ \frac{\sin A}{\cos A} = \tan A \]

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