Question

Use the identity: sin²A + cos²A = 1 to prove that tan²A + 1 = sec²A. Hence, find the value of tan A, when sec A = `5/3`, where A is an acute angle.

Answer 1

`sin^2A + cos^2A = 1`

Dividing both sides by `cos^2A`:

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

`tan^2A + 1 =  sec^2A`

Thus, the identity is proved.

Value of tan A when sec A = `5/3`

So, tan² A + 1 `(5/3)^2`

⇒ tan² A + `25/9 - 1`

⇒ tan² A = `16/9`

⇒ tan A = `sqrt(16/9)`

⇒ tan A = `4/3`