Final answer:
By finding a common denominator and simplifying the expression, we verified that (cos x / (1 - sin x)) + (cos x / (1 + sin x)) = 2 / cos x, simplifies to 2 sec x, confirming option B as the correct answer.
Step-by-step explanation:
To show that (cos x / (1 - sin x)) + (cos x / (1 + sin x)) = 2 sec x, let's start by simplifying the left side of the equation:
- (cos x / (1 - sin x)) + (cos x / (1 + sin x))
- We will find a common denominator, which is (1 - sin x)(1 + sin x) = 1 - sin2x = cos2x.
- Now, we have:
- (cos x * (1 + sin x) + cos x * (1 - sin x)) / cos2x
- Simplify the numerator:
- cos x + cos x * sin x + cos x - cos x * sin x
- The sin x terms cancel out, so we are left with 2 * cos x / cos2x.
- We then can simplify this to 2/cos x.
- Recall that sec x = 1/cos x, so we have 2 * sec x, which matches the right side of the equation.
Therefore, the correct answer is B: (cos x / (1 - sin x)) + (cos x / (1 + sin x)) = 2 / cos x, which simplifies to 2 sec x.