Question

Show that (cos x / (1 - sin x)) + (cos x / (1 + sin x)) = 2 sec x.

A) This is not a valid identity
B) (cos x / (1 - sin x)) + (cos x / (1 + sin x)) = 2 / cos x
C) (cos x / (1 - sin x)) + (cos x / (1 + sin x)) = 2 * sin x
D) (cos x / (1 - sin x)) + (cos x / (1 + sin x)) = 2 / sin x

Answer 1

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.