Question

Simplify: \[ \frac{1}{1+\sin\theta}+\frac{1}{1-\sin\theta}= \]

• A. \(2\cos^2\theta\)
• B. \(-2\cos^2\theta\)
• C. \(2\tan^2\theta\)
• D. \(2\sec^2\theta\)

Hint:

For expressions containing \[ 1\pm \sin\theta, \] multiply or simplify using \[ (1+\sin\theta)(1-\sin\theta)=1-\sin^2\theta=\cos^2\theta. \]

Answer 1

The Correct Option is D

Step 1: Take the LCM of the fractions.
Given, \[ \frac{1}{1+\sin\theta}+\frac{1}{1-\sin\theta} \] Taking LCM, \[ = \frac{(1-\sin\theta)+(1+\sin\theta)} {(1+\sin\theta)(1-\sin\theta)} \]

Step 2: Simplify the numerator.
\[ (1-\sin\theta)+(1+\sin\theta) = 2 \] Thus, \[ = \frac{2} {(1+\sin\theta)(1-\sin\theta)} \]

Step 3: Use the identity \((a+b)(a-b)\).
We know that \[ (a+b)(a-b)=a^2-b^2 \] Hence, \[ (1+\sin\theta)(1-\sin\theta) = 1-\sin^2\theta \] So, \[ = \frac{2}{1-\sin^2\theta} \]

Step 4: Apply the Pythagorean identity.
Using \[ 1-\sin^2\theta=\cos^2\theta, \] we get \[ = \frac{2}{\cos^2\theta} \]

Step 5: Convert into secant form.
Since \[ \sec\theta=\frac{1}{\cos\theta}, \] therefore, \[ \frac{1}{\cos^2\theta}=\sec^2\theta \] Thus, \[ = 2\sec^2\theta \]

Step 6: Match with the options.
The simplified expression is \[ 2\sec^2\theta \]

Step 7: Final conclusion.
Hence, \[ \boxed{2\sec^2\theta} \]