If k + 1 = `sec^2A`(1 - sin A) (1 + sin A), then the value of k is:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( k + 1 = \sec^2 A (1 - \sin A)(1 + \sin A) \), we can follow these steps:

### Step 1: Simplify the Right Side
We start with the expression on the right side:
\[ (1 - \sin A)(1 + \sin A) \]
Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we can simplify this:
\[ (1 - \sin A)(1 + \sin A) = 1^2 - (\sin A)^2 = 1 - \sin^2 A \]

### Step 2: Substitute the Identity
Now we substitute this result back into the equation:
\[ k + 1 = \sec^2 A (1 - \sin^2 A) \]

### Step 3: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ 1 - \sin^2 A = \cos^2 A \]
So we can rewrite the equation as:
\[ k + 1 = \sec^2 A \cdot \cos^2 A \]

### Step 4: Simplify Secant and Cosine
Recall that:
\[ \sec A = \frac{1}{\cos A} \implies \sec^2 A = \frac{1}{\cos^2 A} \]
Thus, we have:
\[ k + 1 = \frac{1}{\cos^2 A} \cdot \cos^2 A \]

### Step 5: Cancel Out Cosine Terms
Now, we can cancel \(\cos^2 A\) from the numerator and the denominator:
\[ k + 1 = 1 \]

### Step 6: Solve for k
To find \(k\), we subtract 1 from both sides:
\[ k = 1 - 1 = 0 \]

### Final Answer
Thus, the value of \(k\) is:
\[ \boxed{0} \]

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