\( \frac { 1 - 2 \sin ^ { 2 } ( \pi + \alpha ) } { \sin ( \frac { \pi } { 2 } + \alpha ) + \sin ( \pi - \alpha ) } + \frac { 1 - 2 \cos ^ { 2 } ( \pi - \alpha ) } { \cos ( \pi + \alpha ) + \cos ( \frac { \pi } { 2 } - \alpha ) } \)

To simplify the expression \[ \frac { 1 - 2 \sin ^ { 2 } ( \pi + \alpha ) } { \sin ( \frac { \pi } { 2 } + \alpha ) + \sin ( \pi - \alpha ) } + \frac { 1 - 2 \cos ^ { 2 } ( \pi - \alpha ) } { \cos ( \pi + \alpha ) + \cos ( \frac { \pi } { 2 } - \alpha ) }, \] let's start with the individual parts. ### First Fraction 1. **Simplifying the numerator**: - We know that \(\sin(\pi + \alpha) = -\sin(\alpha)\). Thus, \[ \sin^2(\pi + \alpha) = \sin^2(\alpha). \] Hence, the numerator becomes: \[ 1 - 2\sin^2(\pi + \alpha) = 1 - 2\sin^2(\alpha). \] 2. **Simplifying the denominator**: - \(\sin\left(\frac{\pi}{2} + \alpha\right) = \cos(\alpha)\) and \(\sin(\pi - \alpha) = \sin(\alpha)\), allowing us to write: \[ \sin\left(\frac{\pi}{2} + \alpha\right) + \sin(\pi - \alpha) = \cos(\alpha) + \sin(\alpha). \] Thus, the first fraction simplifies to: \[ \frac{1 - 2\sin^2(\alpha)}{\cos(\alpha) + \sin(\alpha)}. \] ### Second Fraction 1. **Simplifying the numerator**: - \(\cos(\pi - \alpha) = -\cos(\alpha)\), so \[ \cos^2(\pi - \alpha) = \cos^2(\alpha). \] Therefore, we have: \[ 1 - 2\cos^2(\pi - \alpha) = 1 - 2\cos^2(\alpha). \] 2. **Simplifying the denominator**: - \(\cos(\pi + \alpha) = -\cos(\alpha)\) and \(\cos\left(\frac{\pi}{2} - \alpha\right) = \sin(\alpha)\). This gives us: \[ \cos(\pi + \alpha) + \cos\left(\frac{\pi}{2} - \alpha\right) = -\cos(\alpha) + \sin(\alpha). \] Thus, the second fraction simplifies to: \[ \frac{1 - 2\cos^2(\alpha)}{-\cos(\alpha) + \sin(\alpha)}. \] ### Final Expression Now we combine the simplified fractions: \[ \frac{1 - 2\sin^2(\alpha)}{\cos(\alpha) + \sin(\alpha)} + \frac{1 - 2\cos^2(\alpha)}{-\cos(\alpha) + \sin(\alpha)}. \] Combine these over a common denominator: \[ \frac{(1 - 2\sin^2(\alpha))(-\cos(\alpha) + \sin(\alpha)) + (1 - 2\cos^2(\alpha))(\cos(\alpha) + \sin(\alpha))}{(\cos(\alpha) + \sin(\alpha))(-\cos(\alpha) + \sin(\alpha))}. \] Expanding the numerators: 1. The first part: \[ (1 - 2\sin^2(\alpha))(-\cos(\alpha) + \sin(\alpha)) = -\cos(\alpha) + \sin(\alpha) + 2\sin^2(\alpha)\cos(\alpha) - 2\sin^3(\alpha). \] 2. The second part: \[ (1 - 2\cos^2(\alpha))(\cos(\alpha) + \sin(\alpha)) = \cos(\alpha) + \sin(\alpha) - 2\cos^3(\alpha) - 2\cos^2(\alpha)\sin(\alpha). \] Combine and simplify further: \[ \text{Numerator} = (-\cos(\alpha) + \sin(\alpha)) + (1 - 2\cos^2(\alpha))\cdots \] will depend on values of \( \alpha \), but we can recognize that the expression may further simplify through Pythagorean identities. The full simplification may yield results depending on