Let `x` be the solution set of equation `A^x=Idot,w h e r eA+[0 1-1 4-3 4 3-3 4]a n dI` is the corresponding unit matrix and `xsubeN ,` then the minimum value of `sum(cos^xtheta+sin^xtheta),theta in Rdot`

Answer

2

Explanation

`A=[(0,1,-1),(4,-3,4),(3,-3,4)]`
`implies A^(2)=A.A=[(0,1,-1),(4,-3,4),(3,-3,4)]. [(0,1,-1),(4,-3,4),(3,-3,4)]`
`=[(0+4-3,0-3+3,0+4-4),(0-12+12,4+9-12,-4-12+16),(0-12+12,3+9-12,-3-12+16)]`
`=[(1,0,0),(0,1,0),(0,0,1)]`
`implies A^(2)=I implies A^(4)=A^(6)=A^(8)=I`
Now `A^(x)=I`
`implies x=2, 4, 6, 8`, ...
`:. Sigma (cos^(x) theta + sin^(x) theta)`
`=(cos^(2) theta+sin^(2) theta)+(cos^(4) theta + sin^(4) theta)+(cos^(6) theta + sin^(6) theta)+ `
`=(cos^(2) theta+cos^(4) theta+cos^(6) theta+)+(sin^(4) theta+sin^(4) theta+ sin^(6) theta+)`
`=(cos^(2) theta)/(1- cos^(2) theta)+(sin^(2) theta)/(1-sin^(2) theta)`
`= cot^(2) theta+tan^(2) theta`
Which has minimum value 2