Simplify the expression: (1 / (1 + cosθ)) + (1 / (1 - cosθ)).

Question

Accepted Answer

Step 1: Start with the given expression:
(1 / (1 + cosθ)) + (1 / (1 - cosθ))

Step 2: Find a common denominator:
The common denominator is (1 + cosθ)(1 - cosθ) = 1 - cos²θ.

Step 3: Write the expression as a single fraction:
= [(1 - cosθ) + (1 + cosθ)] / (1 - cos²θ)
= (1 - cosθ + 1 + cosθ) / (1 - cos²θ)
= (2) / (1 - cos²θ)

Step 4: Use the Pythagorean identity:
1 - cos²θ = sin²θ

Step 5: Substitute:
= 2 / sin²θ
= 2cosec²θ

Step 6: Check options:
Option 3 is 2cosec²θ, but the question's options have 2sec²θ as option 2 and 2cosec²θ as option 3.

Re-examining the options, the correct simplification is 2cosec²θ.

Therefore, the correct answer is option 3: 2cosec²θ.

Answer

Answer

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