How do you prove #cscx /(1 + cscx) + cscx /(1 - cscx) = (- 2sinx) / cos^2x#?

1 Answer
May 6, 2015

I would start as:

#(cscx(1-cscx)+cscx(1+cscx))/((1+cscx)(1-cscx))=-2sinx/cos^2x#
#(cscxcancel(-csc^2x)+cscxcancel(+csc^2x))/(1-csc^2x)=-2sinx/cos^2x#
#(2csc(x))/(1-csc^2x)=-2sinx/cos^2x#

I would use the fact that:
#cscx=1/sinx#
So:
#cancel(2)/cancel(sinx)(cancel(sin)^cancel(2)x)/(sin^2x-1)=-cancel(2)cancel(sinx)/cos^2x# collect #-1# on the left side:
#-1/(1-sin^2x)=-1/cos^2x#
But #1-sin^2x=cos^2x#