Show that `x=0` is the only solution satisfying the equation `1+sin^(2) ax=cos x`, where a is irrational.

Explanation

The given equation is
`1+sin^(2) ax = cos x`
or `(1- cos x) + sin^(2) ax=0`
or `2 sin^(2) (x//2)+ sin^(2) ax=0`
Since both the terms on the L.H.S. are non-negative, their sum will be zero only if each one is zero. Thus,
`sin (x//2)=0 and sin ax =0`
`rArr x//2=n pi and ax =m pi, AA n, m in Z`
`rArr x=2n pi and x=m pi//a AA n, m in Z`
`rArr 2 n pi = m pi//a rArr 2 an =m` ...(ii)
Since m and n are integer and a is irrational, Eq. (ii) is possible only if `m=0` and n`=0`.
Hence, `x=0` is the only solution.