Prove that `|cosalpha-cosbeta|lt=|alpha-beta|`

Explanation

`Let alphaltbeta`
We have to fprove that
`alpha-betalecos alpha-cos beta lebeta-alpha`
consider `alpha-betalecos alpha-cos beta`
`therefore cosbeta-beta le cos alpha-alpha`
`f(B)le f(alpha)`,wheref(x) =cosx-x
Now f(x) `=-sinx-1le0, forall` x in R
`f(x)` is decreasing function
`f(beta)lef(alpha)`
`cosbeta-betalecosalpha-alpha`
`alpha-beta le cos alpha-cos beta`
Now consider cos `alpha-cos beta le beta - alpha`
`alpha+cos alpha le beta +cos beta`
`g(alpha)le(beta)`,where g(x) = x+cosx
g(x) SI an increasing function.
`g(alpha)leg(beta)`
`[alpha+cos alpha le beta+cos beta]`
`cos alpha-cos beta le beta -alpha`
From (1) and (2) we get |cos `alpha-cos beta|le|alpha-beta|`