Question

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

Answer 1

`-tan^2x`

Explanation:

Begin from the `color(blue)"basic 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)|)))`

Hence identity simplifies to.

`tan^2x+1=sec^2xrArrtan^2x=sec^2x-1`

multiply through by -1

`rArr-tan^2x=1-sec^2x`