What is the value of k, in the expression, `sec^(2)theta(1+sintheta)(1-sintheta)=k`

Options

A. `1/5`
B. 7
C. 1
D. 12

Answer

To find the value of \( k \) in the expression \( \sec^2 \theta (1 + \sin \theta)(1 - \sin \theta) = k \), we can follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
We know that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we can express \( \sec^2 \theta \) as:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \]
Now substituting this into the expression:
\[ \sec^2 \theta (1 + \sin \theta)(1 - \sin \theta) = \frac{1}{\cos^2 \theta} (1 + \sin \theta)(1 - \sin \theta) \]

### Step 2: Simplify the expression
The expression \( (1 + \sin \theta)(1 - \sin \theta) \) can be simplified using the difference of squares:
\[ (1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta \]
Now substituting this back into the equation gives:
\[ \frac{1}{\cos^2 \theta} (1 - \sin^2 \theta) \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]
Substituting this into the expression, we have:
\[ \frac{1}{\cos^2 \theta} \cdot \cos^2 \theta \]

### Step 4: Cancel out the terms
Now, we can cancel \( \cos^2 \theta \) in the numerator and denominator:
\[ \frac{\cos^2 \theta}{\cos^2 \theta} = 1 \]

### Conclusion
Thus, we find that:
\[ k = 1 \]

### Final Answer
The value of \( k \) is \( 1 \).

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