Find
`(tan A)/((1-cot A))+(cot A)/((1-tan A)) `

Options

A. `(1 + sin A cos A)/(sin A cos A)`
B. `(1+ tan A+cot A)`
C. both `(a)` and `(b)`
D. none of these

Explanation

To solve the problem, we need to simplify the expression:

\[ \frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} \]

### Step 1: Rewrite \(\tan A\) and \(\cot A\) in terms of \(\sin A\) and \(\cos A\)

Recall that:
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A} \]

Substituting these into the expression gives:

\[ \frac{\frac{\sin A}{\cos A}}{1 - \frac{\cos A}{\sin A}} + \frac{\frac{\cos A}{\sin A}}{1 - \frac{\sin A}{\cos A}} \]

### Step 2: Simplify the denominators

For the first term:
\[ 1 - \frac{\cos A}{\sin A} = \frac{\sin A - \cos A}{\sin A} \]
Thus, the first term becomes:
\[ \frac{\frac{\sin A}{\cos A}}{\frac{\sin A - \cos A}{\sin A}} = \frac{\sin^2 A}{\cos A (\sin A - \cos A)} \]

For the second term:
\[ 1 - \frac{\sin A}{\cos A} = \frac{\cos A - \sin A}{\cos A} \]
Thus, the second term becomes:
\[ \frac{\frac{\cos A}{\sin A}}{\frac{\cos A - \sin A}{\cos A}} = \frac{\cos^2 A}{\sin A (\cos A - \sin A)} \]

### Step 3: Combine the two fractions

Now we can combine the two fractions:
\[ \frac{\sin^2 A}{\cos A (\sin A - \cos A)} + \frac{\cos^2 A}{\sin A (\cos A - \sin A)} \]

Notice that \((\cos A - \sin A) = -(\sin A - \cos A)\), so we can rewrite the second term:
\[ \frac{\sin^2 A}{\cos A (\sin A - \cos A)} - \frac{\cos^2 A}{\sin A (\sin A - \cos A)} \]

Now we have a common denominator:
\[ \frac{\sin^2 A \sin A - \cos^2 A \cos A}{\sin A \cos A (\sin A - \cos A)} \]

### Step 4: Factor the numerator

The numerator can be factored:
\[ \sin^3 A - \cos^3 A = (\sin A - \cos A)(\sin^2 A + \sin A \cos A + \cos^2 A) \]

Using the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\), we can simplify:
\[ \sin^2 A + \sin A \cos A + \cos^2 A = 1 + \sin A \cos A \]

### Step 5: Final expression

Thus, the entire expression becomes:
\[ \frac{(\sin A - \cos A)(1 + \sin A \cos A)}{\sin A \cos A (\sin A - \cos A)} \]

The \((\sin A - \cos A)\) terms cancel out:
\[ \frac{1 + \sin A \cos A}{\sin A \cos A} \]

### Step 6: Separate the terms

This can be separated into:
\[ \frac{1}{\sin A \cos A} + 1 \]

### Step 7: Recognize the final result

The term \(\frac{1}{\sin A \cos A}\) can be rewritten using the identity for \(\tan A\) and \(\cot A\):
\[ \frac{1}{\sin A \cos A} = \tan A + \cot A \]

Thus, we have:
\[ 1 + \tan A + \cot A \]

### Conclusion

The left-hand side simplifies to the right-hand side, proving that:
\[ \frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = 1 + \tan A + \cot A \]