How do you solve the identity #(sec(t)+1)(sec(t)-1)=tan^2(t)#?

1 Answer
Bill K.
Sep 15, 2015

You can prove this identity by using the definition of secant and tangent and by using the Pythagorean identity.

Explanation:

Since #cos^2(t)+sin^2(t)=1# for all #t# (this is the Pythagorean identity), it follows that #1+(sin^2(t))/(cos^2(t))=1/(cos^2(t))# for all #t# for which it is defined.

By definition, #tan(t)=(sin(t))/(cos(t))# and #sec(t)=1/(cos(t))#. Therefore, #1+tan^2(t)=sec^2(t)# for all #t# for which it is defined.

Rearranging this equation leads to #sec^2(t)-1=tan^2(t)# and then factoring leads us to conclude that the desired equation is an identity: #(sec(t)+1)(sec(t)-1)=tan^2(t)# for all #t# for which it is defined.

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