Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 208: 20

$$(1+\tan\theta)^2-2\tan\theta=\sec^2\theta$$

$$A=(1+\tan\theta)^2-2\tan\theta$$ $$A=(1+\tan^2\theta+2\tan\theta)-2\tan\theta$$ $$A=1+\tan^2\theta+2\tan\theta-2\tan\theta$$ $$A=1+\tan^2\theta$$ We know from a Pythagorean Identity that $$1+\tan^2\theta=\sec^2\theta$$ That means $$A=\sec^2\theta$$