\( \frac { \sin ^ { 2 } x ( 90 - x ) \cdot \operatorname { con } ( 160 - x ) } { \cos ( 360 ^ { 2 } - x ) \cdot \cos ( 160 - x ) + \sin + x ) } \)

It seems that the expression provided is quite complex, and the calculation is timing out. Let's break it down step by step to simplify it manually. The expression is: \[ \frac { \sin ^ { 2 } x ( 90 - x ) \cdot \cot ( 160 - x ) } { \cos ( 360 ^ { 2 } - x ) \cdot \cos ( 160 - x ) + \sin ( x ) } \] ### Step 1: Rewrite the cotangent Recall that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Therefore, we can rewrite \(\cot(160 - x)\): \[ \cot(160 - x) = \frac{\cos(160 - x)}{\sin(160 - x)} \] ### Step 2: Substitute cotangent in the expression Substituting this into the expression gives: \[ \frac { \sin ^ { 2 } x ( 90 - x ) \cdot \frac{\cos(160 - x)}{\sin(160 - x)} } { \cos ( 360 ^ { 2 } - x ) \cdot \cos ( 160 - x ) + \sin ( x ) } \] ### Step 3: Simplify the numerator The numerator now becomes: \[ \sin^2 x (90 - x) \cdot \cos(160 - x) \] ### Step 4: Rewrite the denominator The denominator remains: \[ \cos(360^2 - x) \cdot \cos(160 - x) + \sin(x) \] ### Step 5: Combine the expression Now we can combine everything: \[ \frac { \sin^2 x (90 - x) \cdot \cos(160 - x) } { \cos(360^2 - x) \cdot \cos(160 - x) + \sin(x) } \] ### Step 6: Analyze the expression At this point, we can analyze the expression further or evaluate it for specific values of \(x\). However, without specific values or further simplification, this is the simplified form of the expression. If you have a specific value for \(x\) or further instructions on how to proceed, please let me know!