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Prove the following:`cos^(-1)((12)/(13))+sin^(-1)(3/5)=sin^(-1)((56)/(65))` |
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Answer» Let `cos^-1(12/13) = x` Then, `cosx = 12/13` `:. sinx = sqrt(1-(12/13)^2) = sqrt(25/169) = 5/13` `:. x = sin^-1(5/13)` `:. L.H.S. = cos^-1(12/13) +sin^-1(3/5) = sin^-1(5/13) +sin^-1(3/5)` We know, `sin^-1A+sin^-1B = sin^-1(Asqrt(1-B^2)+Bsqrt(1-A^2))` So,our expression becomes, `= sin^-1(5/13sqrt(1-(3/5)^2)+3/5sqrt(1-(5/13)^2))` `=sin^-1(5/13*4/5+3/5*12/13)` `=sin^-1(4/13+36/65)` `=sin^-1(56/65) = R.H.S.` |
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