What is the simplified value of
`[(2)/((cotA-tanA))]`?

Options

A. sinA cosA
B. tan2A
C. `tan^(2)A`
D. `sin^(2)A cos^(2)A`

Answer

To simplify the expression \(\frac{2}{\cot A - \tan A}\), we can follow these steps:

### Step 1: Rewrite cotangent and tangent
We know that:
\[ \cot A = \frac{1}{\tan A} \]
Thus, we can rewrite the expression:
\[ \cot A - \tan A = \frac{1}{\tan A} - \tan A \]

### Step 2: Find a common denominator
To combine the terms, we need a common denominator, which is \(\tan A\):
\[ \frac{1}{\tan A} - \tan A = \frac{1 - \tan^2 A}{\tan A} \]

### Step 3: Substitute back into the original expression
Now substitute this back into the original expression:
\[ \frac{2}{\cot A - \tan A} = \frac{2}{\frac{1 - \tan^2 A}{\tan A}} = 2 \cdot \frac{\tan A}{1 - \tan^2 A} \]

### Step 4: Simplify the expression
Now we can simplify:
\[ \frac{2 \tan A}{1 - \tan^2 A} \]

### Step 5: Recognize the identity
We can use the identity \(1 - \tan^2 A = \cos^2 A - \sin^2 A\) to further simplify, but we can also recognize that:
\[ \frac{2 \tan A}{1 - \tan^2 A} = \tan(2A) \]
Thus, we have:
\[ \frac{2 \tan A}{1 - \tan^2 A} = \tan(2A) \]

### Final Answer
Therefore, the simplified value of \(\frac{2}{\cot A - \tan A}\) is:
\[ \tan(2A) \]

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