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Let's go through each trigonometric expression one by one to find their exact values:

### a. \(\sec 60^\circ\)
\[
\sec \theta = \frac{1}{\cos \theta}
\]
Since \(\cos 60^\circ = \frac{1}{2}\),
\[
\sec 60^\circ = \frac{1}{\frac{1}{2}} = 2
\]

### b. \(\csc 45^\circ\)
\[
\csc \theta = \frac{1}{\sin \theta}
\]
Since \(\sin 45^\circ = \frac{\sqrt{2}}{2}\),
\[
\csc 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\]

### c. \(\cot 120^\circ\)
\[
\cot \theta = \frac{1}{\tan \theta}
\]
First, find \(\tan 120^\circ\). Since \(\tan 120^\circ = -\tan 60^\circ = -\sqrt{3}\),
\[
\cot 120^\circ = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
\]

### d. \(\sec 300^\circ\)
\[
\sec 300^\circ = \frac{1}{\cos 300^\circ}
\]
Since \(\cos 300^\circ = \cos (360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2}\),
\[
\sec 300^\circ = \frac{1}{\frac{1}{2}} = 2
\]

### e. \(\csc 135^\circ\)
\[
\csc 135^\circ = \frac{1}{\sin 135^\circ}
\]
Since \(\sin 135^\circ = \sin (180^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}\),
\[
\csc 135^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}
\]

### f. \(\cot 330^\circ\)
\[
\cot 330^\circ = \frac{1}{\tan 330^\circ}
\]
Since \(\tan 330^\circ = \tan (360^\circ - 30^\circ) = -\tan 30^\circ = -\frac{1}{\sqrt{3}}\),
\[
\cot 330^\circ = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3}
\]

### g. \(\sec 150^\circ\)
\[
\sec 150^\circ = \frac{1}{\cos 150^\circ}
\]
Since \(\cos 150^\circ = \cos (180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2}\),
\[
\sec 150^\circ = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}
\]

### h. \(\cot(-30^\circ)\)
\[
\cot(-30^\circ) = \frac{1}{\tan(-30^\circ)}
\]
Since \(\tan(-30^\circ) = -\tan 30^\circ = -\frac{1}{\sqrt{3}}\),
\[
\cot(-30^\circ) = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3}
\]

### Final Answers:
- a. \(\sec 60^\circ = 2\)
- b. \(\csc 45^\circ = \sqrt{2}\)
- c. \(\cot 120^\circ = -\frac{\sqrt{3}}{3}\)
- d. \(\sec 300^\circ = 2\)
- e. \(\csc 135^\circ = \sqrt{2}\)
- f. \(\cot 330^\circ = -\sqrt{3}\)
- g. \(\sec 150^\circ = -\frac{2\sqrt{3}}{3}\)
- h. \(\cot(-30^\circ) = -\sqrt{3}\)

Would you like further details or have any questions?

### Related Questions:
1. What is the exact value of \(\sin 210^\circ\)?
2. How would you compute \(\tan 240^\circ\)?
3. Can you explain why \(\cot 90^\circ\) is undefined?
4. What is the value of \(\sec 270^\circ\) and why?
5. How do you determine which quadrant an angle falls into for trigonometric evaluations?

### Tip:
Remember that trigonometric functions repeat every \(360^\circ\) or \(2\pi\) radians, which can simplify the calculation of angles larger than \(360^\circ\).

Find the exact values of sec 60°, cosec 45°, cot 120°, sec 300°, cosec 135°, cot 330°, sec 150°, and cot(-30°).

Learn how to find the exact values of trigonometric functions for angles such as 60°, 120°, 135°, and others. This detailed solution covers key trigonometric concepts, including secant, cosecant, and cotangent functions.

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