What is the value of `(cot x)/(1-tan x)+ (tan x)/(1-cot x)` ?

Options

A. `sin x cos x +1`
B. `sec x cosec x +1`
C. `tan x cot x +1`
D. `sec^(2)x cosec^(2) x+1`

Answer

To solve the expression \((\cot x)/(1-\tan x) + (\tan x)/(1-\cot x)\), we can follow these steps:

### Step 1: Rewrite cotangent and tangent in terms of sine and cosine
We know that:
\[ \cot x = \frac{\cos x}{\sin x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]
Substituting these into the expression gives:
\[ \frac{\cot x}{1 - \tan x} + \frac{\tan x}{1 - \cot x} = \frac{\frac{\cos x}{\sin x}}{1 - \frac{\sin x}{\cos x}} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\cos x}{\sin x}} \]

### Step 2: Simplify the denominators
For the first term:
\[ 1 - \tan x = 1 - \frac{\sin x}{\cos x} = \frac{\cos x - \sin x}{\cos x} \]
Thus, the first term becomes:
\[ \frac{\frac{\cos x}{\sin x}}{\frac{\cos x - \sin x}{\cos x}} = \frac{\cos^2 x}{\sin x (\cos x - \sin x)} \]

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

### Step 3: Combine the two fractions
Now we can combine the two fractions:
\[ \frac{\cos^2 x}{\sin x (\cos x - \sin x)} + \frac{\sin^2 x}{\cos x (\sin x - \cos x)} \]
Notice that \((\sin x - \cos x) = -(\cos x - \sin x)\), thus we can rewrite the second term:
\[ \frac{\sin^2 x}{\cos x (-1)(\cos x - \sin x)} = -\frac{\sin^2 x}{\cos x (\cos x - \sin x)} \]
Now we can combine them:
\[ \frac{\cos^2 x - \sin^2 x}{(\cos x - \sin x)(\sin x \cos x)} \]

### Step 4: Factor the numerator
The numerator can be factored using the difference of squares:
\[ \cos^2 x - \sin^2 x = (\cos x - \sin x)(\cos x + \sin x) \]
Thus, we have:
\[ \frac{(\cos x - \sin x)(\cos x + \sin x)}{(\cos x - \sin x)(\sin x \cos x)} \]

### Step 5: Cancel common factors
Assuming \(\cos x \neq \sin x\), we can cancel \((\cos x - \sin x)\):
\[ \frac{\cos x + \sin x}{\sin x \cos x} \]

### Step 6: Final expression
Thus, the final simplified expression is:
\[ \frac{\cos x + \sin x}{\sin x \cos x} \]