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If F(x) =[(cos x, -sin x,0),(sin x,cos x...

If `F(x) =[(cos x, -sin x,0),(sin x,cos x,0),(0,0,1)]` show that F(x)F(y)=F(x+y)`

Text Solution

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Here, `F(x) = [[cosx,-sinx,0],[sinx,cosx,0],[0,0,1]]`
`:. F(y) = [[cosy,-siny,0],[siny,cosy,0],[0,0,1]]`
`:.F(x+y) = [[cos(x+y),-sin(x+y),0],[sin(x+y),cos(x+y),0],[0,0,1]]`
Now,`L.H.S. = F(x)*F(y)`
`= [[cosx,-sinx,0],[sinx,cosx,0],[0,0,1]] [[cosy,-siny,0],[siny,cosy,0],[0,0,1]]`
`=[[cosxcosy-sinxsiny,-cosxsiny-sinxsiny,0],[sinxcosy+cosxsiny,-sinxsiny+cosxcosy,0],[0,0,1]]`
`=[[cos(x+y),-sin(x+y),0],[sin(x+y),cos(x+y),0],[0,0,1]]`
`=F(x+y) = R.H.S.`
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