Given : F(x) = \( \begin{bmatrix} cos\,x&-sin\,x &0\\[0.3em] sin\,x&cos\,x &0\\[0.3em] 0 & 0 & 1 \end{bmatrix}\).
To show : F(x) . F(y) = F(x + y).
Formula used :
If A is a matrix of order a x b and B is a matrix of order c x d , then matrix AB exists and is of order a x d ,
if and only if b = c
We know that,
cosx(cosy) – sinx (siny)
= cos(x+y) and -cosx(siny) - sinx(cosy)
= -sin(x+y)
F(x + y) = F(x) . F(y)