Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 208: 19

$$(1+\sin t)^2+\cos^2t=2\sin t+2$$

$$A=(1+\sin t)^2+\cos^2t$$ $$A=(1+\sin^2t+2\sin t)+\cos^2 t$$ $$A=1+(\sin^2 t+\cos^2 t)+2\sin t$$ Following one Pythagorean Identity: $$\sin^2 t+\cos^2 t=1$$ we have $$A=1+1+2\sin t$$ $$A=2\sin t+2$$