Show that (cosec A - sin A) (sec A - cos A) = 1 / (tan A + cot A).

To prove this equation true we will use trigonometric identities. We will start from LHS.

Answer: Hence Proved (cosec A - sin A) (sec A - cos A) = 1 / tan A + cot A.

Let's see the detailed solution.

Explanation:

LHS = (cosec A - sin A) (sec A - cos A)

cosec A can be written as 1/sin A and sec A can be written as 1 / cos A

(1 / sin A - sin A) (1 / cos A - cos A)

On solving this expression using 1 - sin²A = cos²A and sin²A = 1 - cos²A, we get

[(1 - sin²A) / sin A] [(1 - cos²A) / cos A]

(cos²A / sin A) (sin²A / cos A)

sin A × cos A / 1

By using sin²A + cos²A = 1

sin A × cos A / (sin²A + cos²A)

On dividing both numerator and denominator by cos A sin A, we get,

1 / (sin²A + cos²A) sin A × cos A

On solving this, we get

1 / (sin²A/ sin A × cos A) + (cos²A / sin A × cos A)

1 / (sin A / cos A) + (cos A / sin A)

1/ tan A + cot A = RHS

Hence Proved that LHS = RHS