Question

How do you simplify the expression `sec^2x-1`?

Answer 1

`tan^2x`

Explanation:

Using the `color(blue)"trigonometric identity"`

`color(red)(bar(ul(|color(white)(a/a)color(black)(sin^2x+cos^2x=1)color(white)(a/a)|)))`

divide all terms on both sides by `cos^2x`

`rArr(sin^2x)/(cos^2x)+(cos^2x)/(cos^2x)=1/(cos^2x)`

`color(orange)"Reminder"`

`color(red)(bar(ul(|color(white)(a/a)color(black)(tanx=(sinx)/(cosx)" and " secx=1/(cosx))color(white)(a/a)|)))`

`rArrtan^2x+1=sec^2x`

subtract 1 from both sides.

`tan^2xcancel(+1)cancel(-1)=sec^2x-1`

`rArrsec^2x-1=tan^2x`