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प्रश्न
Prove the following result:
`sin^-1 12/13+cos^-1 4/5+tan^-1 63/16=pi`
उत्तर
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
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