Find the value of `((1)/(costheta)+(1)/(cottheta))((1)/(costheta)-(1)/(cottheta))`.

Options

A. 0
B. `-1`
C. 1
D. 2

Answer

To find the value of the expression \(\left(\frac{1}{\cos \theta} + \frac{1}{\cot \theta}\right)\left(\frac{1}{\cos \theta} - \frac{1}{\cot \theta}\right)\), we can follow these steps:

### Step 1: Rewrite the expression
Let:
\[ a = \frac{1}{\cos \theta} \quad \text{and} \quad b = \frac{1}{\cot \theta} \]
Thus, the expression can be rewritten as:
\[ (a + b)(a - b) \]

### Step 2: Apply the difference of squares identity
Using the identity \( (a + b)(a - b) = a^2 - b^2 \), we can simplify the expression:
\[ a^2 - b^2 \]

### Step 3: Calculate \(a^2\)
Now, we calculate \(a^2\):
\[ a^2 = \left(\frac{1}{\cos \theta}\right)^2 = \frac{1}{\cos^2 \theta} \]

### Step 4: Calculate \(b^2\)
Next, we calculate \(b^2\):
\[ b = \frac{1}{\cot \theta} = \frac{1}{\frac{\cos \theta}{\sin \theta}} = \frac{\sin \theta}{\cos \theta} \]
Thus,
\[ b^2 = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta} \]

### Step 5: Substitute \(a^2\) and \(b^2\) back into the expression
Now substitute \(a^2\) and \(b^2\) into the expression:
\[ a^2 - b^2 = \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} \]

### Step 6: Combine the fractions
Since both terms have the same denominator, we can combine them:
\[ \frac{1 - \sin^2 \theta}{\cos^2 \theta} \]

### Step 7: Simplify using the Pythagorean identity
Using the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\), we have:
\[ \frac{\cos^2 \theta}{\cos^2 \theta} = 1 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{1} \]