Question

How do you verify `(1+csc(x))/(cot(x)+cos(x)) = sec(x)`?

Answer 1

On the left side,

`(1+cscx)/(cotx+cosx)`

Rewrite `cscx` and `cotx` in terms of `sinx` and `cosx`.

`=(1+color(blue)(1/(sinx)))/(color(purple)((cosx)/(sinx))+cosx)`

Simplify.

`=((sinx+1)/(sinx))/((cosx+sinxcosx)/(sinx))`

`=(sinx+1)/(sinx)*(sinx)/(cosx+sinxcosx)`

`=(sinx+1)/(color(red)cancelcolor(black)(sinx))*(color(red)cancelcolor(black)(sinx))/(cosx+sinxcosx)`

`=(sinx+1)/(cosx+sinxcosx)`

Factor out `cosx` from the denominator.

`=(sinx+1)/(cosx(1+sinx))`

Simplify.

`=color(red)cancelcolor(black)(sinx+1)/(cosxcolor(red)cancelcolor(black)((1+sinx)))`

`=1/cosx`

`=secx`