Evaluate the value of the expression:

$$\sin 25^\circ \sin 65^\circ - \cos 25^\circ \cos 65^\circ$$

Question

Evaluate the value of the expression:

$$\sin 25^\circ \sin 65^\circ - \cos 25^\circ \cos 65^\circ$$

Accepted Answer

**Solution:**

1. Recall the complementary angle identities:
   - $$\sin(90^\circ - a) = \cos a$$
   - $$\cos(90^\circ - a) = \sin a$$

2. Given expression:
   $$\sin 25^\circ \sin 65^\circ - \cos 25^\circ \cos 65^\circ$$

3. Substitute $$65^\circ = 90^\circ - 25^\circ$$:
   $$\sin 25^\circ \sin (90^\circ - 25^\circ) - \cos 25^\circ \cos (90^\circ - 25^\circ)$$

4. Using the identities:
   $$\sin 25^\circ \cos 25^\circ - \cos 25^\circ \sin 25^\circ$$

5. Simplify:
   $$\sin 25^\circ \cos 25^\circ - \cos 25^\circ \sin 25^\circ = 0$$

**Answer:** $0$

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