Sum Difference Formulas 5E23B0

1. **Problem Statement:** We need to solve a trigonometric expression or equation using only the sum and difference formulas, without using Simpson's rule or double and half-angle formulas. 2. **Sum and Difference Formulas:** The sum and difference formulas for sine and cosine are: $$\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$$ $$\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$$ These formulas allow us to express the sine or cosine of sums or differences of angles in terms of sines and cosines of individual angles. 3. **Approach:** Since the problem does not specify the exact expression, the key is to rewrite any trigonometric expression involving sums or differences of angles using these formulas only. 4. **Example:** Suppose we want to find $$\sin 75^\circ$$ using sum formulas: $$\sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$$ Using known values: $$\sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 30^\circ = \frac{1}{2}$$ So: $$\sin 75^\circ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$$ 5. **Summary:** Use only the sum and difference formulas to rewrite and solve trigonometric expressions. Since the user did not provide a specific expression, this is the general method to solve such problems using sum and difference formulas only.