Question

Verify that each equation is an identity?

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

Answer 1

Multiply out the brackets and then apply the Pythagorean Identity `Tan^2(x) + 1 = Sec^2(x)`

Explanation:

Left Side = (SecA - TanA)(SecA + TanA)
= `Sec^2A + SecATanA - TanASecA - Tan^2A`

Notice that `SecATanA - TanASecA = 0`

So Left Side = `Sec^2A - Tan^2A`

Now apply the Pythagorean Identity `Tan^2A + 1 = Sec^2A` by replacing the `Sec^2A` by `Tan^2A + 1`

Left Side = `Tan^2A + 1 - Tan^2A`

Left Side = 1

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