(i) Find the maximum and minimum values of `sin^(4)x + cos^(2)x` and hence or otherwise find the maximum value of `sin^(1000)x + cos^(2008)x`. (ii) Find the maximum value of cos (cos x)

Explanation

(i) We have,
`sin^(4)x + cos^(2)x=sin^(4)x+1-sin^(2)x`
`=(sin^(2)x)^(2)-2.sin^(2)x.1/2+1/4+3/4`
`=3/4+(1/2-sin^(2)x)^(2) ge 3/4` for all `x in R`
and also, `sin^(4)x + cos^(2)x le 3/4+ 1/4=1 AA x int R`
Hence, `3/4 le sin^(4)x+cos^(2)x le 1, AA x int R`
`rArr "min"(sin^(4)x+cos^(2)x)=3/4` which occurs at `x=+-pi/4`
and `"max"(sin^(4)x+cos^(2)x)=1`. which occurs at `x=npi`.
(ii) We have,
`-1 le cos x le 1`
`rArr cos1 le cos(cos x) le 1`
`rArr cos1 le cos(cos (cos x)) le cos(cos 1)`
Hence, maximum value `=cos(cos1)`