Question

Which expression is equivalent to sec^2x − 1? a. cot2x b. tan2x c. csc2x d. cos2x

Answer 1

option B tan²x

Explanation:

We have to simplify the expression given as (
`sec^(2)`x-1 )

Since sec x =
`(1)/(cosx)`

so (
`(1)/(cosx)` )² - 1 =
`(1)/(cos^(2) x)` - 1

=
`(1-cos^(2)x )/(cos^(2)x)`

=
`(sin^(2)x )/(cos^(2)x)`

= tan²x

Therefore, option B tan²x is the answer.

Answer 2

Sec^2 x - 1 = tan^2x

Proof:
Sec^2x = 1+ tan^2x

1/cos^2x = 1 + sin^2x/cos^2x

1/cos^2x - sin^2x/cos^2x = 1
Using common denominator:
(1-sin^2x)/cos^2x = 1
sin^2x + cos^2 x = 1
cos^2 x = 1 - sin^2x
Substituting :
cos^2x/cos^2x = 1
1 = 1
Left hand side = right hand side