`(1)/(tan 3 A - tan A)- (1)/(cot 3 A - cot A)=`

Options

A. `tan A `
B. `tan 2A`
C. `cot A `
D. `cot 2A `

Answer

To solve the expression \(\frac{1}{\tan 3A - \tan A} - \frac{1}{\cot 3A - \cot A}\), we will simplify it step by step.

### Step 1: Rewrite the expression
The first step is to rewrite the expression using the definitions of tangent and cotangent:
\[ \frac{1}{\tan 3A - \tan A} - \frac{1}{\cot 3A - \cot A} \]
can be rewritten as:
\[ \frac{1}{\frac{\sin 3A}{\cos 3A} - \frac{\sin A}{\cos A}} - \frac{1}{\frac{\cos 3A}{\sin 3A} - \frac{\cos A}{\sin A}} \]

### Step 2: Find a common denominator
For the first term, the common denominator is \(\cos 3A \cos A\):
\[ \frac{\cos A - \sin A \cdot \tan 3A}{\cos 3A \cos A} \]
For the second term, the common denominator is \(\sin 3A \sin A\):
\[ \frac{\sin A - \cos A \cdot \cot 3A}{\sin 3A \sin A} \]

### Step 3: Simplify each fraction
Now we can simplify each fraction:
1. For the first term:
\[ \frac{\cos A \sin 3A - \sin A \cos 3A}{\sin 3A \cos A \cos 3A} \]
This can be simplified using the sine difference identity:
\[ \sin(3A - A) = \sin 2A \]
So, it becomes:
\[ \frac{\sin 2A}{\sin 3A \cos A \cos 3A} \]

2. For the second term:
\[ \frac{\sin A \cos 3A - \cos A \sin 3A}{\sin 3A \sin A \sin A} \]
This can also be simplified using the sine difference identity:
\[ \sin(3A - A) = \sin 2A \]
So, it becomes:
\[ \frac{\sin 2A}{\sin 3A \sin^2 A} \]

### Step 4: Combine the fractions
Now we can combine the two fractions:
\[ \frac{\sin 2A}{\sin 3A \cos A \cos 3A} - \frac{\sin 2A}{\sin 3A \sin^2 A} \]
Finding a common denominator:
\[ \frac{\sin 2A (\sin^2 A - \cos A \cos 3A)}{\sin 3A \cos A \cos 3A \sin^2 A} \]

### Step 5: Final simplification
The expression simplifies to:
\[ \frac{\sin 2A (\sin^2 A - \cos A \cos 3A)}{\sin 3A \cos A \cos 3A \sin^2 A} \]

### Final Answer:
Thus, the final answer is:
\[ \frac{\sin 2A (\sin^2 A - \cos A \cos 3A)}{\sin 3A \cos A \cos 3A \sin^2 A} \]