Question

सिद्ध कीजिए कि -
`cos^(-1)((cos alpha+cos beta)/(1+cos alpha cos beta))=2tan^(-1)(tan.(alpha)/(2)tan.(beta)/(2)).`

Answer 1

R.H.S. `=2tan^(-1)(tan.(alpha)/(2)tan.(beta)/(2))`
`rArr" R.H.S. "=cos^(-1){(1-tan^(2).(alpha)/(2)tan^(2).(beta)/(2))/(1+tan^(2).(alpha)/(2)tan^(2).(beta)/(2))},`
`[because 2tan^(-1)x=cos^(-1)((1-x^(2))/(1=x^(2)))]`
`=cos^(-1){(cos^(2).(alpha)/(2)cos^(2).(beta)/(2)-sin^(2).(alpha)/(2)sin^(2).(beta)/(2))/(cos^(2).(alpha)/(2)cos^(2).(beta)/(2)-sin^(2).(alpha)/(2)sin^(2).(beta)/(2))}`
`=cos^(-1){((2cos^(2).(alpha)/(2))(2cos^(2).(beta)/(2))-(2sin^(2).(alpha)/(2))(2sin^(2).(beta)/(2)))/((2cos^(2).(alpha)/(2))(2cos^(2).(beta)/(2))+(2sin^(2).(alpha)/(2))(2sin^(2).(beta)/(2)))}`
`=cos^(-1){((1+cos alpha)(1+cos beta)-(1-cos alpha)(1-cos beta))/((1+cos alpha)(1+cos beta)+(1-cos alpha)(1-cos beta))}`
`=cos^(-1)((cos alpha +cos beta)/(1+cos alpha cos beta))`
= L.H.S.
यही सिद्ध करना था ।