How do you find the exact value of the expression: `sin(2*tan^-1(5/12))`

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The exact value of `sin(2*tan^-1(5/12))` has to be determined.

Let `y = tan^-1(5/12)`

=> `tan y = 5/12`

`1 + tan^2y = sec^2y = 1 + (5/12)^2`

=> `sec^2y = 169/144`

=> `sec y = 13/12`

=> `cos y = 12/13`

=> `y = cos^-1(12/13)`

`sin y = sqrt(1 - (12/13)^2)`

=> `sin y = 5/13`

=> `y = sin^-1(5/13)`

`sin(2*tan^-1(5/12))`

=> `sin(2*sin^-1(5/13))`

=> `2*sin(sin^-1(5/13))*cos(cos^-1(12/13))`

=> `2*(5/13)*(12/13)`

=> `120/169`

The value of `sin(2*tan^-1(5/12))` = `120/169`

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