Question

Use the identity sin² A + cos² A = 1 to prove that tan² A + 1 = sec² A. Hence, find the value of tan A when sec A = `5/3`, where A is an acute angle.

Answer 1

sin² A + cos² A = 1

Divided by cos² A.

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

tan² A + 1 = sec² A

Hence proved.

Given, sec A = `5/3`

sec² A = `25/9`  ...(By squaring)

tan² A + 1 = `25/9`  ....(∵ sec² A = tan² A + 1)

⇒ tan² A = `25/9 - 1`

⇒ tan² A = `(25 - 9)/9`

⇒ tan² A = `16/9`

⇒ tan A = `±4/3`

But tan A ≠ `(-4)/3` ...(A is an acute angle.)

So, tan A = `4/3`