Question 391597
{{{x = det(matrix(2,2,p, cos A,q,-sin A))/ det(matrix(2,2,sin A, cos A, cos A,-sin A))}}}= q cos A - p sin A

{{{y = det(matrix(2,2,sin A, p, cos A,q))/ det(matrix(2,2,sin A, cos A, cos A,-sin A))}}}= p cos A - q sin A

after using Cramer's rule to solve for x and y.  
Then 
{{{x^2 + y^2}}}
={{{q^2(cos A)^2 - 2pq sin A cos A + p^2(sin A )^2 + p^2(cos A)^2 - 2pq sin A cos A + q^2(sin A )^2}}}
={{{(p^2 + q^2)(cos A)^2  + (p^2 + q^2)(sin A)^2 - 4pq sin A cos A }}}
={{{p^2 + q^2 - 2pq sin 2A}}}.