Chapter 6 - Trigonometric Functions - 6.3 Properties of the Trigonometric Functions - 6.3 Assess Your Understanding - Page 396: 131

$(\sin\theta \cos\phi)^2+(\sin\theta \sin\phi)^2+(\cos\theta)^2=\sin\theta^2\cos\phi^2+\sin\theta^2\sin\phi^2+(\cos\theta)^2=(\sin\theta)^2(\cos\phi^2+\sin\phi^2)+(\cos\theta)^2$ I know that $\cos\phi^2+\sin\phi^2=1$, hence: $(\sin\theta)^2(\cos\phi^2+\sin\phi^2)+(\cos\theta)^2=(\sin\theta)^2\cdot1+(\cos\theta)^2=(\sin\theta)^2+(\cos\theta)^2=1.$

$(\sin\theta \cos\phi)^2+(\sin\theta \sin\phi)^2+(\cos\theta)^2=\sin\theta^2\cos\phi^2+\sin\theta^2\sin\phi^2+(\cos\theta)^2=(\sin\theta)^2(\cos\phi^2+\sin\phi^2)+(\cos\theta)^2$ I know that $\cos\phi^2+\sin\phi^2=1$, hence: $(\sin\theta)^2(\cos\phi^2+\sin\phi^2)+(\cos\theta)^2=(\sin\theta)^2\cdot1+(\cos\theta)^2=(\sin\theta)^2+(\cos\theta)^2=1.$