`((cosectheta-sectheta)(cottheta-tantheta))/((cosectheta+sectheta)(sectheta.cosectheta-2))=?`

Options

A. a) 2
B. b) 1
C. c) 3
D. d) `-1`

Answer

To solve the expression \(\frac{(\csc \theta - \sec \theta)(\cot \theta - \tan \theta)}{(\csc \theta + \sec \theta)(\sec \theta \csc \theta - 2)}\), we will follow these steps:

### Step 1: Rewrite the Trigonometric Functions
We start by rewriting the trigonometric functions in terms of sine and cosine:
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Substituting these into the expression gives:
\[ \frac{\left(\frac{1}{\sin \theta} - \frac{1}{\cos \theta}\right)\left(\frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta}\right)}{\left(\frac{1}{\sin \theta} + \frac{1}{\cos \theta}\right)\left(\frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} - 2\right)} \]

### Step 2: Simplify the Numerator
Now, simplify the numerator:
\[ \frac{1}{\sin \theta} - \frac{1}{\cos \theta} = \frac{\cos \theta - \sin \theta}{\sin \theta \cos \theta} \]
\[ \frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta} = \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta} \]
Thus, the numerator becomes:
\[ \frac{(\cos \theta - \sin \theta)(\cos^2 \theta - \sin^2 \theta)}{\sin^2 \theta \cos^2 \theta} \]

### Step 3: Simplify the Denominator
Now simplify the denominator:
\[ \frac{1}{\sin \theta} + \frac{1}{\cos \theta} = \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta} \]
\[ \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} - 2 = \frac{1}{\sin \theta \cos \theta} - 2 \]
This can be rewritten as:
\[ \frac{1 - 2\sin \theta \cos \theta}{\sin \theta \cos \theta} \]
Thus, the denominator becomes:
\[ \frac{(\sin \theta + \cos \theta)(1 - 2\sin \theta \cos \theta)}{\sin^2 \theta \cos^2 \theta} \]

### Step 4: Combine the Numerator and Denominator
Now we can combine the numerator and denominator:
\[ \frac{(\cos \theta - \sin \theta)(\cos^2 \theta - \sin^2 \theta)}{(\sin \theta + \cos \theta)(1 - 2\sin \theta \cos \theta)} \]

### Step 5: Factor and Simplify
Notice that \(\cos^2 \theta - \sin^2 \theta\) can be factored as \((\cos \theta - \sin \theta)(\cos \theta + \sin \theta)\):
\[ \frac{(\cos \theta - \sin \theta)^2(\cos \theta + \sin \theta)}{(\sin \theta + \cos \theta)(1 - 2\sin \theta \cos \theta)} \]
The \((\sin \theta + \cos \theta)\) terms cancel out:
\[ \frac{(\cos \theta - \sin \theta)^2}{1 - 2\sin \theta \cos \theta} \]

### Step 6: Use Pythagorean Identity
Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), we can simplify further:
\[ (\cos \theta - \sin \theta)^2 = \cos^2 \theta + \sin^2 \theta - 2\sin \theta \cos \theta = 1 - 2\sin \theta \cos \theta \]
Thus, we have:
\[ \frac{1 - 2\sin \theta \cos \theta}{1 - 2\sin \theta \cos \theta} = 1 \]

### Final Answer
Therefore, the final answer is:
\[ \boxed{1} \]