**Solution:**

1/(secA - tanA)

= (secA + tanA)/[(secA + tanA)(secA - tanA)]

= (secA + tanA)/(sec^{2}A - tan^{2}A)

= secA + tanA

because sec^{2}A - tan^{2}A = 1.

Proof:

sin^{2}A + cos^{2}A = 1

sin^{2}A/cos^{2}A + 1 = 1/cos^{2}A

tan^{2}A + 1 = sec^{2}A

1 = sec^{2}A - tan^{2}A