Question

How do you simplify `(sin^2(t)+cos^2(t) )/ cos^2(t)`?

Answer 1

`sec^2t`

Explanation:

Divide each term on the numerator by `cos^2t`

`rArr(sin^2t)/cos^2t+(cos^2t)/cos^2t=tan^2t+1........ (A)`

From the identity `color(red)(|bar(ul(color(white)(a/a)color(black)(sin^2t+cos^2t=1)color(white)(a/a)|)))`
divide each term by `cos^2t`

`(sin^2t)/cos^2t+(cos^2t)/cos^2t=1/cos^2t` we obtain the identity.

`color(red)(|bar(ul(color(white)(a/a)color(black)(tan^2t+1=sec^2t)color(white)(a/a)|)))`
Substitute this result into the right side of (A)

`rArr(sin^2t+cos^2t)/cos^2t=sec^2t`