InterviewSolution
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InterviewSolution
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Show that `sin^(-1)(12)/(13)+cos^(-1)4/5+tan^(-1)(63)/(16)=pi`. |
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Answer» Let, `sin^-1(12/13) = x and cos^-1(4/5) = y and tan^-1(63/16)=z->(1)` Then,`sin x = 12/13 and cos y = 4/5` `tan x = 5/12 and tan y = 3/4` `x = tan^-1(5/12) and y = tan^-1(3/4)->(2)` From (1) and (2), `sin^-1(12/13) + cos^-1(4/5) = tan^-1(12/5)+tan^-1(3/4) ` We know, `tan^-1x+tan-^-1y = tan^-1((x+y)/(1-xy))` `tan^-1(12/5)+tan^-1(3/4) = tan^-1((12/5+3/4)/(1-12/5**3/4))` `=tan^-1((63/20)/(-4/5)) = tan^-1(-63/16)->(3)` From (1),`tan^-1(63/16) = z`, So, `tan z = 63/16` `=>-tan z = -63/16` `=>tan(pi - z) = -63/16` `=>tan^-1(-63/16) = pi-z->(4)` So, From(1), (3) and (4), `L.H.S. = pi-z+z = pi = R.H.S.` |
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