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Prove that `2tan^(-1)(1/2)+tan^(-1)(1/7)=sin^(-1)((31)/(25sqrt(2)))` |
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Answer» `L.H.S. = 2tan^-1(1/2)+tan^-1(1/7)` `=tan^-1((2*1/2)/(1-(1/2)^2))+tan^-1(1/7)` `=tan^-1((1)/(3/4))+tan^-1(1/7)` `=tan^-1(4/3)+tan^-1(1/7)` `=tan^-1((4/3+1/7)/(1-(4/3)(1/7)))` `=tan^-1((31/21)/(17/21))` `=tan^-1(31/17)` Let `tan^-1(31/17) = theta` Then, `tantheta = 31/17` `cot theta = 17/31` `cosectheta = sqrt(1+(17/31)^2) = sqrt1250/31 = (25sqrt2)/31` `:. sintheta = 31/(25sqrt2)` `=>theta = sin^-1(31/(25sqrt2))` `:. tan^-1(31/17) = sin^-1(31/(25sqrt2)) = R.H.S.` |
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