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If y=log (1+ sin x), prove that y(4)+y(...

If `y=log (1+ sin x)`, prove that `y_(4)+y_(3)y_(1)+y_(2)^(2)=0`.

Text Solution

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`y=log (1+ sin x)" (1)"`
`y_(1)=(cos x)/(1+ sin x)" (2)"`
`y_(2)=(-sin x (1+ sin x)-cos x cos x)/((1+sin x)^(2))`
`=(-(1+ sin x))/(1+ sin x)^(2)`
`=-(1)/((1+ sin x))" (3)"`
`y_(3)=(cos x)/((1+ sin x)^(2))=(cos x)/(1+ sin x)xx(1)/(1+ sin x)=-y_(1)y_(2)" (4)"`
`therefore" "y_(4)=-y_(2)^(2)-y_(1)y_(3)`
`"or " y_(4)+y_(3)y_(1)+y_(2)^(2)=0`
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