1. Evaluate the expression: $\csc 10^\circ – \sqrt{3} \sec 10^\circ$
a) 8
b) 6
c) 4
d) 2
Answer: (c) 4
Solution: Convert the terms into sine and cosine:
$$\frac{1}{\sin 10^\circ} – \frac{\sqrt{3}}{\cos 10^\circ} = \frac{\cos 10^\circ – \sqrt{3} \sin 10^\circ}{\sin 10^\circ \cos 10^\circ}$$
Multiply and divide the numerator by 2:
$$\frac{2\left(\frac{1}{2}\cos 10^\circ – \frac{\sqrt{3}}{2}\sin 10^\circ\right)}{\sin 10^\circ \cos 10^\circ} = \frac{2(\sin 30^\circ \cos 10^\circ – \cos 30^\circ \sin 10^\circ)}{\sin 10^\circ \cos 10^\circ}$$
Apply the identity $\sin(A-B)$:
$$\frac{2\sin(30^\circ – 10^\circ)}{\sin 10^\circ \cos 10^\circ} = \frac{2\sin 20^\circ}{\frac{1}{2}\sin 20^\circ} = 4$$