Question

The value of (tan A cosec A)² − (sin A sec A)² is ______.

Options

• 0

• 1

• −1

• 2

Answer 1

The value of (tan A cosec A)² − (sin A sec A)² is 1.

Explanation:

Given: (tan A cosec A)² − (sin A sec A)²

`tan A = (sin A)/(cos A), "cosec" A = 1/(sin A), sec A = 1/(cos A)`

`tan A  "cosec A" = (sin A)/(cos A) xx 1/(sin A) = 1/(cos A)`

`sin A sec A = sin A xx 1/(cos A) = (sin A)/(cos A) = tan A`

Squaring both:

`(tan A  "cosec" A)^2 = (1/(cos A))^2 = 1/(cos^2 A)`

`(sin A sec A)^2 = (tan A)^2 = (sin^2 A)/(cos^2 A)`

`1/cos^2 A - (sin^2 A)/(cos^2 A)`

`(1 - sin^2 A)/(cos^2 A)`

Using the identity 1 − sin²A = cos²A, we get

`(cos^2 A)/(cos^2 A) = 1`