\( ( \frac { \cos \beta } { \sin \alpha } + \frac { \sin \beta } { \cos \alpha } ) \cdot \frac { 1 - \cos 4 \alpha } { \cos ( \pi - \beta + \alpha ) } \)

To simplify the expression \[ ( \frac { \cos \beta } { \sin \alpha } + \frac { \sin \beta } { \cos \alpha } ) \cdot \frac { 1 - \cos 4 \alpha } { \cos ( \pi - \beta + \alpha ) }, \] we will break it down into parts. 1. **Simplifying the first part:** The first part of the expression is \[ \frac { \cos \beta } { \sin \alpha } + \frac { \sin \beta } { \cos \alpha }. \] We can combine these fractions over a common denominator: \[ = \frac{\cos \beta \cos \alpha + \sin \beta \sin \alpha}{\sin \alpha \cos \alpha}. \] Using the angle addition formula for cosine, we have: \[ \cos(\beta - \alpha) = \cos \beta \cos \alpha + \sin \beta \sin \alpha. \] Thus, we can rewrite the expression as: \[ = \frac{\cos(\beta - \alpha)}{\sin \alpha \cos \alpha}. \] 2. **Simplifying the second part:** The second part of the expression is \[ \frac { 1 - \cos 4 \alpha } { \cos ( \pi - \beta + \alpha ) }. \] We know that \[ \cos(\pi - x) = -\cos x, \] so we can rewrite \[ \cos(\pi - \beta + \alpha) = -\cos(\beta - \alpha). \] Therefore, the second part becomes: \[ \frac{1 - \cos 4 \alpha}{-\cos(\beta - \alpha)} = -\frac{1 - \cos 4 \alpha}{\cos(\beta - \alpha)}. \] 3. **Combining both parts:** Now we can combine both parts of the expression: \[ \left( \frac{\cos(\beta - \alpha)}{\sin \alpha \cos \alpha} \right) \cdot \left( -\frac{1 - \cos 4 \alpha}{\cos(\beta - \alpha)} \right). \] The \(\cos(\beta - \alpha)\) terms cancel out, leading to: \[ = -\frac{1 - \cos 4 \alpha}{\sin \alpha \cos \alpha}. \] 4. **Final expression:** Thus, the simplified expression is: \[ -\frac{1 - \cos 4 \alpha}{\sin \alpha \cos \alpha}. \] This is the final simplified form of the original expression.