Question

if F(x)=[cosx−sinx0sinxcosx0001], show that F(x)F(y)=F(x+y).

Answer 1

Given :  F(x) = $\left[\begin{array}{ccc}Cosx&-Sinx&0\\Sinx&Cosx&0\\0&0&1\end{array}\right]$

To find :  Show that F(x)F(y) = F(x + y)

Solution:

F(x )  =  $\left[\begin{array}{ccc}Cosx&-Sinx&0\\Sinx&Cosx&0\\0&0&1\end{array}\right]$

F(y)  =  $\left[\begin{array}{ccc}Cosy&-Siny&0\\Siny&Cosy&0\\0&0&1\end{array}\right]$

F(x) * F(y)  = $\left[\begin{array}{ccc}CosxCosy - SinxSiny&-CosxSiny - SinxCosy&0\\SinxCosy + CosxSiny&-SinxSiny + CosxCosy&0\\0&0&1\end{array}\right]$

Cos(x + y) = CosxCosy - Sinxy

Sin(x + y) = SinxCosy + CosxSiny

F(x) * F(y) = $\left[\begin{array}{ccc}Cosx(x + y)&-Sin(x + y)&0\\Sin(x + y)&Cos(x + y) &0\\0&0&1\end{array}\right]$

=> F(x) * F(y) = F(x + y)

QED

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