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

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Answer 1 · 2015-12-13T17:19:22

The Pythagorean identity

#sin^2x+cos^2x=1#

can be divided by #cos^2x# to see that

#tan^2x+1=sec^2x#

Then, #1# can be subtracted from either side to see that

#tan^2x=sec^2x-1#

So, the identity on the left can be rewritten as

#(sec^2x-1)/(secx-1)#

Now, see that #sec^2x-1# is a difference of squares which can be factored into

#(secx+1)(secx-1)#

Substitute this into the expression to get

#((secx+1)(secx-1))/(secx-1)#

Which simplifies to be

#(secx+1)#