(c) $$ \cos ^{2} A\left[\frac{1}{\sin A-1}+\frac{1}{\sin A+1}\right]=-2 \sin A $$

Question

UpStudy AI · Accepted Answer

We want to show that

cos²A [1/(sinA – 1) + 1/(sinA + 1)] = –2 sinA.

Step 1. Simplify the expression inside the brackets:

1/(sinA – 1) + 1/(sinA + 1)

To add the fractions, combine them over a common denominator:

= [ (sinA + 1) + (sinA – 1) ] / [(sinA – 1)(sinA + 1)]
  = (2 sinA) / (sin²A – 1)

Step 2. Recall the Pythagorean identity:

sin²A + cos²A = 1  ⟹  sin²A – 1 = –cos²A

Thus, the expression becomes:

= (2 sinA) / (–cos²A)
  = –(2 sinA) / cos²A

Step 3. Multiply by cos²A (as in the original expression):

cos²A * [–(2 sinA)/cos²A] = –2 sinA

This confirms that

cos²A [1/(sinA – 1) + 1/(sinA + 1)] = –2 sinA.

Note: The expression is valid provided that sinA ≠ 1 and sinA ≠ –1 (to avoid division by zero).
cos²A [1/(sinA – 1) + 1/(sinA + 1)] equals –2 sinA.

(c) \( \cos ^{2} A\left[\frac{1}{\sin A-1}+\frac{1}{\sin A+1}\right]=-2 \sin A \)