Prove the following: `cos^(-1)((12)/(13))+sin^(-1)(3/5)=sin^(-1)((56)/(65))`

Explanation

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))`

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