Let P=(cosA sinA)
(sinA cosA)
Q=(cosB sinA)
(sinB cosA) show that
PQ=(cosA(A-B) sin(A B))
(sinA(A B) cos(A-B))
To show that PQ = (cos(A-B) sin(A+B))
(sin(A-B) cos(A+B))
First, let's multiply the matrices P and Q together:
PQ = P * Q
To multiply matrices, we need to multiply corresponding entries and sum them up. Using the given matrices P and Q, the multiplication is as follows:
PQ = ((cosA * cosB + sinA * sinB) (cosA * sinA + sinA * cosA))
((sinA * cosB + cosA * sinB) (sinA * sinB + cosA * cosB))
Now, let's simplify the individual entries of PQ:
PQ = (cos(A-B) sin(A-B))
(sin(A-B) cos(A-B))
To simplify further, we can use the trigonometric identity:
cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B)
Substituting these identities into the matrix PQ:
PQ = (cos(A) * cos(B) + sin(A) * sin(B) sin(A) * cos(B) - cos(A) * sin(B))
(sin(A) * cos(B) - cos(A) * sin(B) cos(A) * cos(B) + sin(A) * sin(B))
Simplifying this expression gives us the final result:
PQ = (cos(A-B) sin(A+B))
(sin(A-B) cos(A+B))
Hence, we have shown that PQ is equal to (cos(A-B) sin(A+B))
(sin(A-B) cos(A+B)).