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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)

Evaluate the following trigonometric expressions:

(a) sin(765°)
(b) cos(-60°)
(c) cosec(-1410°)
(d) sec(240°)
(e) tan(780°)
(f) sin(-225°)
(g) cot(210°)
(h) sec(135°)

Learn to evaluate trigonometric expressions involving large or negative angles, including sin(765°), cos(-60°), and sec(240°). This problem demonstrates how to use angle reduction and periodic properties in trigonometry. Suitable for high school students in Grades 10-12.

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