`((1-tantheta)/(1-cottheta))^(2)+1=`

Options

A. `cos^(2) theta`
B. `sin^(2) theta`
C. `cosec^(2) theta`
D. `sec^(2) theta`

Answer

To solve the expression \(\left(\frac{1 - \tan \theta}{1 - \cot \theta}\right)^2 + 1\), we will follow these steps:

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

Recall that:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

Substituting these into the expression gives us:

\[ \left(\frac{1 - \frac{\sin \theta}{\cos \theta}}{1 - \frac{\cos \theta}{\sin \theta}}\right)^2 + 1 \]

### Step 2: Simplify the fractions

Now, we need to simplify the numerator and the denominator:

**Numerator:**
\[ 1 - \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta - \sin \theta}{\cos \theta} \]

**Denominator:**
\[ 1 - \frac{\cos \theta}{\sin \theta} = \frac{\sin \theta - \cos \theta}{\sin \theta} \]

### Step 3: Substitute back into the expression

Now substituting these back into our expression:

\[ \left(\frac{\frac{\cos \theta - \sin \theta}{\cos \theta}}{\frac{\sin \theta - \cos \theta}{\sin \theta}}\right)^2 + 1 \]

### Step 4: Simplify the complex fraction

This can be simplified as follows:

\[ \left(\frac{\cos \theta - \sin \theta}{\sin \theta - \cos \theta} \cdot \frac{\sin \theta}{\cos \theta}\right)^2 + 1 \]

Notice that \(\sin \theta - \cos \theta = -(\cos \theta - \sin \theta)\), so we can rewrite the fraction:

\[ \left(-\frac{\sin \theta}{\cos \theta}\right)^2 + 1 \]

### Step 5: Simplify further

This simplifies to:

\[ \left(\frac{\sin \theta}{\cos \theta}\right)^2 + 1 = \tan^2 \theta + 1 \]

### Step 6: Use the Pythagorean identity

Using the identity \(1 + \tan^2 \theta = \sec^2 \theta\):

\[ \tan^2 \theta + 1 = \sec^2 \theta \]

### Final Answer

Thus, the final answer is:

\[ \sec^2 \theta \]

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