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