How do you verify #(1 + sec^2 x) / (1 + tan^2 x) = 1 + cos^2 x#?

1 Answer
Jul 3, 2016

Use the Pythagorean Identity #1+tan^2x=sec^2x#.

Explanation:

Recall the Pythagorean Identity #1+tan^2x=sec^2x# (this can be derived by dividing the identity #sin^2x+cos^2x=1# by #cos^2x#). The key to this problem is applying this identity.

Since #1+tan^2x=sec^2x#, we can replace the #1+tan^2x# in the denominator with #sec^2x#:
#(1+sec^2x)/(sec^2x)=1+cos^2x#

Now we can break the fraction up in two:
#1/sec^2x+sec^2x/sec^2x=1+cos^2x#

Since #1/secx=cosx#, #1/sec^2x=cos^2x#; and #sec^2x/sec^2x=1#. So:
#cos^2x+1=1+cos^2x#

Using the commutative property of addition we can rearrange the left side of the equation to match the right:
#1+cos^2x=1+cos^2x#