Simplify the Matrix

Problem

[[cos(x)*sin(x),−y*sin(x)*sin(x),y*cos(x)*cos(x)],[sin(x)*sin(x),y*cos(x)*sin(x),y*sin(x)*cos(x)],[cos(x),0,−y*sin(x)]]

Solution

1. Identify the matrix elements and look for common factors within columns or rows to simplify the notation.

2. Apply trigonometric identities where possible. Note that sin(x)*sin(x)=sin2(x) and cos(x)*cos(x)=cos2(x)

3. Rewrite the matrix using power notation for the trigonometric functions to make the expression more concise.

4. Factor out the variable y from the second and third columns if representing the matrix as a product, though standard simplification usually involves writing the entries in their simplest functional form.

Final Answer

[[cos(x)*sin(x),−y*sin2(x),y*cos2(x)],[sin2(x),y*cos(x)*sin(x),y*sin(x)*cos(x)],[cos(x),0,−y*sin(x)]]=[[cos(x)*sin(x),−y*sin2(x),y*cos2(x)],[sin2(x),y*cos(x)*sin(x),y*sin(x)*cos(x)],[cos(x),0,−y*sin(x)]]

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