Question

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

Answer 1

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.