InterviewSolution
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InterviewSolution
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Prove that:`sin^(-1){(sqrt(1+x)+sqrt(1-x))/2}=pi/4+(sin^(-1)x)/2,""0 < x < 1` |
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Answer» `L.H.S. = sin^-1((sqrt(1+x)+sqrt(1-x))/2)` Let `x = sintheta => theta = sin^-1x` Then, `L.H.S. = sin^-1((sqrt(1+sintheta)+sqrt(1-sintheta))/2)` `= sin^-1((sqrt(1+cos(pi/2-theta))+sqrt(1-cos(pi/2-theta)))/2)` `= sin^-1((sqrt(2cos^2(pi/4-theta/2))+sqrt(2sin^2(pi/4-theta/2)))/2)` `= sin^-1((sqrt2cos(pi/4-theta/2)+sqrt2sin(pi/4-theta/2))/2)` `= sin^-1(1/sqrt2cos(pi/4-theta/2)+1/sqrt2sin(pi/4-theta/2))` `= sin^-1(sin(pi/4)cos(pi/4-theta/2)+cos(pi/4)sin(pi/4-theta/2))` `= sin^-1(sin(pi/4+pi/4-theta/2))` `= sin^-1(sin(pi/2-theta/2))` `=pi/2-theta/2` `=pi/2-sin^-1(x)/2` `:. sin^-1((sqrt(1+x)+sqrt(1-x))/2) = pi/2-sin^-1(x)/2` |
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