Question

Prove the following result:

`sin^-1  12/13+cos^-1  4/5+tan^-1  63/16=pi`

Answer 1

LHS = `sin^-1  12/13+cos^-1  4/5+tan^-1  63/16`

`=tan^-1  (12/13)/sqrt(1-144/169)+tan^-1  sqrt(1-16/25)/(4/5)+tan^-1  63/16`     `[becausesin^-1x=tan^-1  x/sqrt(1-x^2)   and   cos^-1x=tan^-1   sqrt(1-x^2)/x]`

`=tan^-1  (12/13)/(5/13)+tan^-1  (3/5)/(4/5)+tan^-1  63/16`

`=tan^-1  12/5+tabn^-1  3/4+tan^-1  63/16`

`=pi+tan^-1((12/5+3/4)/(1-12/5xx3/4))+tan^-1  63/16`       `[because tan^-1x+tan^-1y=pi+tan^-1((x+y)/(1-xy))]`

`=pi+tan^-1((63/20)/(-16/20))+tan^-1  63/16`

`=pi+tan^-1  (-63)/16+tan^-1  63/16`

`=pi-tan^-1  63/16+tan^-1  63/16`
= π = RHS