Write everything in terms of #sin(x)# and #cos(x)# using the fact that #sec(x)# and #csc(x)# are reciprocals of #sin(x)# and #cos(x)#, respectively:
#sec^2(x)=1/cos^2(x)#
#csc^2(x)=1/sin^2(x)#
#1/cos^2(x) - 1/(cos^2(x)sin^2(x))#
Subtract. We'll use #cos^2(x)sin^2(x)# as a common denominator, meaning we multiply #1/cos^2(x)# by #sin^2(x)/sin^2(x)#, the equivalent of multiplying by #1#. .
#sin^2(x)/(cos^2(x)sin^2(x)) - 1/(cos^2(x)sin^2(x))#
#(sin^2(x)-1)/(cos^2(x)sin^2(x))#
Recall this identity:
#sin^2(x)+cos^2(x)=1#
So:
#sin^2(x)-1=-cos^2(x)#
We can now rewrite by replacing our numerator with #-cos^2(x)# :
#(-cos^2(x))/(cos^2(x)sin^2(x))#
#cos^2(x) # cancels out:
#(-cancelcos^2(x))/(cancelcos^2(x)sin^2(x))#
#(-1)/(sin^2(x))#
#=-csc^2(x)#
