(0) \( \frac{\cos ^{2}\left(180^{\circ}-\theta\right)-\cos ^{2} \cdot \alpha-\sin ^{2}\left(\alpha-180^{\circ}\right)}{\cos \left(360^{\circ}-2 \theta\right)-1} \)

Simplify the expression by following steps: - step0: Solution: \(\sin\left(\left(alpha-180\right)^{2}\right)\) - step1: Multiply the terms: \(\sin\left(\left(a^{2}lph-180\right)^{2}\right)\) - step2: Expand the expression: \(\sin\left(a^{4}l^{2}p^{2}h^{2}-360a^{2}lph+32400\right)\) Calculate or simplify the expression \( \cos(alpha)^2 \). Simplify the expression by following steps: - step0: Solution: \(\cos\left(\left(alpha\right)^{2}\right)\) - step1: Multiply the terms: \(\cos\left(\left(a^{2}lph\right)^{2}\right)\) - step2: Calculate: \(\cos\left(a^{4}l^{2}p^{2}h^{2}\right)\) Calculate or simplify the expression \( \cos(360 - 2 * \theta) \). Simplify the expression by following steps: - step0: Solution: \(\cos\left(360-2\theta \right)\) Calculate or simplify the expression \( \cos(180 - \theta)^2 \). Simplify the expression by following steps: - step0: Solution: \(\cos\left(\left(180-\theta \right)^{2}\right)\) - step1: Expand the expression: \(\cos\left(32400-360\theta +\theta ^{2}\right)\) Calculate or simplify the expression \( \cos(2 * \theta - 360) - 1 \). Simplify the expression by following steps: - step0: Solution: \(\cos\left(2\theta -360\right)-1\) Let's simplify the expression step by step. The original expression is: \[ \frac{\cos ^{2}\left(180^{\circ}-\theta\right)-\cos ^{2} \cdot \alpha-\sin ^{2}\left(\alpha-180^{\circ}\right)}{\cos \left(360^{\circ}-2 \theta\right)-1} \] ### Step 1: Simplify the Numerator 1. **Calculate \( \cos(180^\circ - \theta)^2 \)**: \[ \cos(180^\circ - \theta) = -\cos(\theta) \implies \cos^2(180^\circ - \theta) = \cos^2(\theta) \] 2. **The term \( \cos^2(\alpha) \)** remains as is. 3. **Calculate \( \sin(\alpha - 180^\circ)^2 \)**: \[ \sin(\alpha - 180^\circ) = -\sin(\alpha) \implies \sin^2(\alpha - 180^\circ) = \sin^2(\alpha) \] Now, substituting these results into the numerator: \[ \cos^2(\theta) - \cos^2(\alpha) - \sin^2(\alpha) \] ### Step 2: Combine the Terms in the Numerator Using the identity \( \sin^2(\alpha) + \cos^2(\alpha) = 1 \): \[ \cos^2(\theta) - (\cos^2(\alpha) + \sin^2(\alpha)) = \cos^2(\theta) - 1 \] ### Step 3: Simplify the Denominator 1. **Calculate \( \cos(360^\circ - 2\theta) \)**: \[ \cos(360^\circ - 2\theta) = \cos(2\theta) \] Thus, the denominator becomes: \[ \cos(2\theta) - 1 \] ### Step 4: Final Expression Now, substituting the simplified numerator and denominator back into the expression: \[ \frac{\cos^2(\theta) - 1}{\cos(2\theta) - 1} \] ### Step 5: Further Simplification Using the identity \( \cos^2(\theta) - 1 = -\sin^2(\theta) \): \[ \frac{-\sin^2(\theta)}{\cos(2\theta) - 1} \] Using the identity \( \cos(2\theta) = 1 - 2\sin^2(\theta) \): \[ \cos(2\theta) - 1 = -2\sin^2(\theta) \] Thus, the expression simplifies to: \[ \frac{-\sin^2(\theta)}{-2\sin^2(\theta)} = \frac{1}{2} \] ### Final Answer The simplified expression is: \[ \frac{1}{2} \]