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

1 Answer
May 21, 2016

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