InterviewSolution
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InterviewSolution
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`cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=(x)/(2), x in (0,(pi)/(4))` |
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Answer» We have `(1+sinx)={cos^(2)(x//2)+sin^(2)(x//2)+2sin(x//2)cos(x//2)}` `={cos(x//2)+sin(x//2)}^(2).` `(1-sinx)={cos^(2)(x//2)+sin^(2)(x//2)-2sin(x//2)cos(x//2)}` `={cos(x//2)-sin(x//2)}^(2).` `thereforesqrt(1+sinx)=sqrt({cos(x//2)+sin(x//2)}^(2))=cos((x)/(2))+sin((x)/(2))` `andsqrt(1-sinx)=sqrt({cos(x//2)-sin(x//2)}^(2))=cos((x)/(2))-sin((x)/(2)).` `thereforey=cot^(-1){([cos(x//2)+sin(x//2)]+[cos(x//2)-sin(x//2)])/([cos(x//2)+sin(x//2)]-[cos(x//2)-sin(x//2)])}` `=cot^(-1){(2cos(x//2))/(2sin(x//2))}=cot^(-1){cot(x//2)}=(x)/(2)` `rArr(dy)/(dx)=(d)/(dx)((x)/(2))=(1)/(2).` |
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