K = sen(90°₊×)₊cos(270°₊×)₊sen(180° - x)₊cos(360° - x)?

Bueno lo que debemos aplicar son las propiedades de suma y resta de senos, además suma y resta de cosenos.

Suma de senos

<img src="https://tex.z-dn.net/?f=sin%28x%2By%29%3Dsin%28x%29cos%28y%29%2Bsin%28y%29cos%28x%29" />

Resta de senos

<img src="https://tex.z-dn.net/?f=sin%28x-y%29%3Dsin%28x%29cos%28y%29-sin%28y%29cos%28x%29" />

Suma de cosenos

<img src="https://tex.z-dn.net/?f=cos%28x%2By%29%3Dcos%28x%29cos%28y%29-sin%28x%29sin%28y%29" />

Resta de cosenos

<img src="https://tex.z-dn.net/?f=cos%28x-y%29%3Dcos%28x%29cos%28y%29%2Bsin%28x%29sin%28y%29" />

Además :

<img src="https://tex.z-dn.net/?f=sin%2890%20%5E%7Bo%7D%20%29%3D1%20%5C%5C%20cos%2890%5E%7Bo%7D%29%3D0%20%5C%5C%20sin%28270%5E%7Bo%7D%29%3D-1%20%5C%5C%20cos%28270%5E%7Bo%7D%29%3D0%20%5C%5C%20sin%28180%5E%7Bo%7D%29%3D0%20%5C%5C%20cos%28180%5E%7Bo%7D%29%3D-1%20%5C%5C%20sin%28360%5E%7Bo%7D%29%3D0%20%5C%5C%20cos%28360%5E%7Bo%7D%29%3D1" />

Con todo ésto vamos a resolver el ejercicio

<img src="https://tex.z-dn.net/?f=K%3Dsin%2890%5E%7Bo%7D%2Bx%29%2Bcos%28270%5E%7Bo%7D%2Bx%29%2Bsin%28180%5E%7Bo%7D-x%29%2Bcos%28360%5E%7Bo%7D-x%29%20%5C%5C%20%20%5C%5C%20%5B%5D%20%5Bsin%2890%29cos%28x%29%2Bsin%28x%29cos%2890%29%5D%2B%5Bcos%28270%29cos%28x%29-sin%28270%29sin%28x%29%5D%2B..%20%5C%5C%20sin%28180%29cos%28x%29-sin%28x%29cos%28180%29%2B%5Bcos%28360%29cos%28x%29%2Bsin%28360%29sin%28x%29%5D%20%20%5C%5C%20%20%5C%5C%20%28cos%28x%29%29%2B%280cos%28x%29-%28-1%29sin%28x%29%29%2B%280cos%28x%29-sin%28x%29%28-1%29%29%2B...%20%5C%5C%20...%2B%28%281%29cos%28x%29%2B0sin%28x%29%29%20%5C%5C%20%20%5C%5C%20cos%28x%29%2Bsin%28x%29%2Bsin%28x%29%2Bcos%28x%29%3D2cos%28x%29%2B2sin%28x%29" />

Y eso sería todo.

Espero te sirva y si tienes alguna pregunta.