# Verify that each equation is an identity?

## (sec a - tan a)(sec a + tan a) = 1

Nov 29, 2016

Multiply out the brackets and then apply the Pythagorean Identity $T a {n}^{2} \left(x\right) + 1 = S e {c}^{2} \left(x\right)$

#### Explanation:

Left Side = (SecA - TanA)(SecA + TanA)
= $S e {c}^{2} A + S e c A T a n A - T a n A S e c A - T a {n}^{2} A$

Notice that $S e c A T a n A - T a n A S e c A = 0$

So Left Side = $S e {c}^{2} A - T a {n}^{2} A$

Now apply the Pythagorean Identity $T a {n}^{2} A + 1 = S e {c}^{2} A$ by replacing the $S e {c}^{2} A$ by $T a {n}^{2} A + 1$

Left Side = $T a {n}^{2} A + 1 - T a {n}^{2} A$

Left Side = 1

Left Side = Right Side [Q.E.D.]