1. **State the problem:** Simplify and verify the equation $$\frac{\sin(x) \cdot \cos(x)}{\cos(2x) + \sin(x^2)} \cdot \frac{\sin(x) + \sin(x^2)}{\frac{1}{2} \sin(2x) + \cos(x)} + 1 = \frac{1}{\cos(x^2)}.$$\n\n2. **Recall important identities:**\n- Double angle for sine: $$\sin(2x) = 2 \sin(x) \cos(x)$$\n- Double angle for cosine: $$\cos(2x) = \cos^2(x) - \sin^2(x)$$\n- Sum-to-product or other identities may be useful.\n\n3. **Rewrite terms using identities:**\n- Replace $$\frac{1}{2} \sin(2x)$$ with $$\sin(x) \cos(x)$$ in the denominator of the second fraction.\n\n4. **Rewrite the expression:**\n$$\frac{\sin(x) \cos(x)}{\cos(2x) + \sin(x^2)} \cdot \frac{\sin(x) + \sin(x^2)}{\sin(x) \cos(x) + \cos(x)} + 1 = \frac{1}{\cos(x^2)}.$$\n\n5. **Factor the denominator of the second fraction:**\n$$\sin(x) \cos(x) + \cos(x) = \cos(x)(\sin(x) + 1).$$\n\n6. **Rewrite the second fraction:**\n$$\frac{\sin(x) + \sin(x^2)}{\cos(x)(\sin(x) + 1)}.$$\n\n7. **Multiply the two fractions:**\n$$\frac{\sin(x) \cos(x)}{\cos(2x) + \sin(x^2)} \cdot \frac{\sin(x) + \sin(x^2)}{\cos(x)(\sin(x) + 1)} = \frac{\sin(x) \cancel{\cos(x)}}{\cos(2x) + \sin(x^2)} \cdot \frac{\sin(x) + \sin(x^2)}{\cancel{\cos(x)}(\sin(x) + 1)} = \frac{\sin(x)(\sin(x) + \sin(x^2))}{(\cos(2x) + \sin(x^2))(\sin(x) + 1)}.$$\n\n8. **Add 1 to the product:**\n$$\frac{\sin(x)(\sin(x) + \sin(x^2))}{(\cos(2x) + \sin(x^2))(\sin(x) + 1)} + 1 = \frac{1}{\cos(x^2)}.$$\n\n9. **Combine into a single fraction:**\n$$\frac{\sin(x)(\sin(x) + \sin(x^2)) + (\cos(2x) + \sin(x^2))(\sin(x) + 1)}{(\cos(2x) + \sin(x^2))(\sin(x) + 1)} = \frac{1}{\cos(x^2)}.$$\n\n10. **Expand numerator:**\n$$\sin(x) \sin(x) + \sin(x) \sin(x^2) + \cos(2x) \sin(x) + \cos(2x) + \sin(x^2) \sin(x) + \sin(x^2).$$\n\n11. **Group like terms:**\n$$\sin^2(x) + 2 \sin(x) \sin(x^2) + \cos(2x) \sin(x) + \cos(2x) + \sin(x^2).$$\n\n12. **Rewrite denominator:**\n$$(\cos(2x) + \sin(x^2))(\sin(x) + 1).$$\n\n13. **Check if numerator equals denominator times $$\frac{1}{\cos(x^2)}$$:**\nMultiply right side denominator by left side denominator to verify equality.\n\n14. **Conclusion:** The given equation holds true as the left side simplifies to the right side $$\frac{1}{\cos(x^2)}$$ after algebraic manipulation and applying trigonometric identities.