1. **Problem statement:** Calculate the value of $$\sin^2 35^\circ + \sin^2 10^\circ + \sqrt{2} \sin 35^\circ \sin 10^\circ$$. 2. **Formula and rules:** Recall the identity for sine squared and product of sines. We can use the cosine of difference formula and Pythagorean identities to simplify. 3. **Step 1:** Use the identity $$\sin^2 x = \frac{1 - \cos 2x}{2}$$ to rewrite the sine squares: $$\sin^2 35^\circ = \frac{1 - \cos 70^\circ}{2}, \quad \sin^2 10^\circ = \frac{1 - \cos 20^\circ}{2}$$ 4. **Step 2:** Substitute these into the expression: $$\frac{1 - \cos 70^\circ}{2} + \frac{1 - \cos 20^\circ}{2} + \sqrt{2} \sin 35^\circ \sin 10^\circ$$ 5. **Step 3:** Combine the fractions: $$\frac{2 - \cos 70^\circ - \cos 20^\circ}{2} + \sqrt{2} \sin 35^\circ \sin 10^\circ$$ 6. **Step 4:** Use the product-to-sum formula for $$\sin A \sin B$$: $$\sin A \sin B = \frac{\cos(A-B) - \cos(A+B)}{2}$$ So, $$\sin 35^\circ \sin 10^\circ = \frac{\cos 25^\circ - \cos 45^\circ}{2}$$ 7. **Step 5:** Substitute back: $$\frac{2 - \cos 70^\circ - \cos 20^\circ}{2} + \sqrt{2} \times \frac{\cos 25^\circ - \cos 45^\circ}{2}$$ 8. **Step 6:** Multiply and combine: $$= \frac{2 - \cos 70^\circ - \cos 20^\circ}{2} + \frac{\sqrt{2}}{2} (\cos 25^\circ - \cos 45^\circ)$$ 9. **Step 7:** Use known cosine values: $$\cos 70^\circ \approx 0.3420, \quad \cos 20^\circ \approx 0.9397, \quad \cos 25^\circ \approx 0.9063, \quad \cos 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071$$ 10. **Step 8:** Calculate numeric values: $$\frac{2 - 0.3420 - 0.9397}{2} = \frac{0.7183}{2} = 0.35915$$ $$\frac{\sqrt{2}}{2} (0.9063 - 0.7071) = 0.7071 \times 0.1992 = 0.1408$$ 11. **Step 9:** Sum the parts: $$0.35915 + 0.1408 = 0.49995 \approx 0.5$$ 12. **Final answer:** The value is approximately $$\frac{1}{2}$$. **Answer choice:** (5) 1/2