#### MATH 1150

##### Final - Practice 1 | Fall '11| Muir
For each equation either verify that the equation is an identity, or find a counterexample to show that it is not an identity.

a. $\tan^{2}2x+\sin^{2}2x+\cos^{2}2x=\sec^{2}2x$

b. $\sin(3\phi) = 3\sin\phi−4\sin^{3}\phi$

c. $\cos(\alpha+\beta)+cos(\alpha−\beta)= 2\cos\alpha\cos\beta$

d. $\cos\theta−\sin\theta = \pm\sqrt{ \cos(2\theta)}$

e.

f. $\tan(\frac{z}{2})=\frac{\tan z}{\sec z+1}$

2. Find all real solutions.

a. $\tan\alpha = 1$

b. $2\sin\beta+\sqrt{3} = 0$

c. $\sin 2\omega = \cos\omega$

d. $\sin y + \cos y = 1$

e. $2\cos^{2}z+3\cos z = −1$ with $0 ≤ z \lt 2π$

f. $\sin\phi + \tan\phi− \cos\phi −1 = 0$ with $0 ≤ \phi \lt 2π$

g. $\tan^{2}θ−6\tan θ=−8$ with $0≤θ<2π$

h. $5\sin t − 3\cos t + \cos t \sin t−15 = 0$
i. $2\arcsin x = \frac{π}{2}$
j. $\sin ρ≥{\frac{1}{2}}$