Let's evaluate each trigonometric expression by reducing the angles to their equivalent within the standard range $[0^\circ, 360^\circ]$.

Solution:

1. (a) $\sin(765^\circ)$:

   • $765^\circ \mod 360^\circ = 45^\circ$
   • $\sin(765^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$

2. (b) $\cos(-60^\circ)$:

   • $-60^\circ + 360^\circ = 300^\circ$
   • $\cos(-60^\circ) = \cos(300^\circ) = \frac{1}{2}$

3. (c) $\csc(-1410^\circ)$:

   • $-1410^\circ + 4 \times 360^\circ = -1410 + 1440 = 30^\circ$
   • $\csc(-1410^\circ) = \csc(30^\circ) = 2$

4. (d) $\sec(240^\circ)$:

   • $\sec(240^\circ) = \sec(240^\circ) = -2$ (since $\sec(240^\circ) = \frac{1}{\cos(240^\circ)} = -2$)

5. (e) $\tan(780^\circ)$:

   • $780^\circ \mod 360^\circ = 60^\circ$
   • $\tan(780^\circ) = \tan(60^\circ) = \sqrt{3}$

6. (f) $\sin(-225^\circ)$:

   • $-225^\circ + 360^\circ = 135^\circ$
   • $\sin(-225^\circ) = \sin(135^\circ) = \frac{\sqrt{2}}{2}$

7. (g) $\cot(210^\circ)$:

   • $\cot(210^\circ) = \cot(210^\circ) = \frac{1}{\tan(210^\circ)} = \frac{1}{\tan(30^\circ)} = \sqrt{3}$

8. (h) $\sec(135^\circ)$:

   • ( \sec(135^\circ) = -\sec(135^\circ)