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Prove that`cot^(-1)((sqrt(1+sin)+sqrt(1-sinx))/(1sqrt(1+sin)-sqrt(1-sinx)))=x/2; x in (0,pi/4)dot` |
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Answer» `sqrt(1+sinx) = sqrt(1+2sin(x/2)cos(x/2))` (As `sin2theta = 2sinthetacostheta`) `=sqrt(cos^2(x/2)+sin^2(x/2)+2sin(x/2)cos(x/2))` (As `cos^2theta+sin^2theta = 1`) `=sqrt(cos(x/2)+sin(x/2)^2)` `:. sqrt(1+sinx) =cos(x/2)+sin(x/2)` Now, `sqrt(1-sinx) = sqrt(1-2sin(x/2)cos(x/2))` (As `sin2theta = 2sinthetacostheta`) `=sqrt(cos^2(x/2)+sin^2(x/2)-2sin(x/2)cos(x/2))` `=sqrt(cos(x/2)-sin(x/2)^2)` `:.sqrt(1-sinx) =cos(x/2)-sin(x/2)` So, `L.H.S. = cot^-1[(sqrt(1+sinx)+sqrt(1-sinx) )/(sqrt(1+sinx)-sqrt(1-sinx))]` `= 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 = R.H.S.` |
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