1. **Problem:** Simplify the expression $\sin^4\theta + \cos^4\theta$ and find its equivalent form. 2. Start by recognizing the expression is a sum of fourth powers of sine and cosine. 3. Use the identity $(a^2+b^2)^2 = a^4 + 2a^2b^2 + b^4$ to relate the terms. Here, let $a = \sin^2\theta$ and $b = \cos^2\theta$. 4. Thus, $$ (\sin^2\theta + \cos^2\theta)^2 = \sin^4\theta + 2\sin^2\theta\cos^2\theta + \cos^4\theta $$ 5. Using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, substitute: $$ 1^2 = \sin^4\theta + 2\sin^2\theta\cos^2\theta + \cos^4\theta $$ 6. Rearranged to isolate the original expression: $$ \sin^4\theta + \cos^4\theta = 1 - 2\sin^2\theta\cos^2\theta $$ 7. Further, use the double-angle identity $\sin 2\theta = 2\sin\theta\cos\theta$, hence: $$ \sin^2 2\theta = (2\sin\theta\cos\theta)^2 = 4\sin^2\theta\cos^2\theta $$ 8. Substitute back: $$ \sin^4\theta + \cos^4\theta = 1 - \frac{1}{2}\sin^2 2\theta $$ **Final equivalent expression:** $$ \boxed{\sin^4\theta + \cos^4\theta = 1 - \frac{1}{2}\sin^2 2\theta} $$