Question

How do you simplify the expression `tan^2x/(secx+1)`?

Answer 1

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

`rArr tan^2 x/(sec x+1)=sec x-1, or, =(1-cos x)/cos x`,

Answer 2

`secx-1`

Explanation:

Use the `color(blue)"trigonometric identity"` for `tan^2x`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(tan^2x=sec^2x-1)color(white)(a/a)|)))`

`rArr(tan^2x)/(secx+1)=(sec^2x-1)/(secx+1)........ (A)`

Note that `sec^2x-1" is a difference of squares"` and in general, factorises as.

`color(red)(|bar(ul(color(white)(a/a)color(black)(a^2-b^2=(a-b)(a+b))color(white)(a/a)|)))`

`(secx)^2=sec^2x" and " (1)^2=1rArra=secx" and " b=1`

`rArrsec^2x-1=(secx-1)(secx+1)`

substitute into (A)

`rArr(sec^2x-1)/(secx+1)=((secx-1)cancel((secx+1)))/cancel(secx+1)`

`rArr(tan^2x)/(secx+1)=secx-1`