हल करे :
`cos^(-1) xsqrt(3) + cos^(-1) x = (pi)/(2)`

Explanation

दिया है, `cos^(-1) xsqrt(3) +cos^(-1) x = (pi)/(2)`
या `cos^(-1) x sqrt(3) = (pi)/(2) - cos^(-1)x` या `xsqrt(3) = cos((pi)/(2) -cos^(-1) x)`
या `xsqrt(3) = sin( cos^(-1)x)" "[because cos((pi)/(2) -theta)= sin theta]" "...(2)`
माना कि `cos^(-1) x = theta,` तो `x = cos theta`
अब `sin^(2) theta = 1 - cos^(2) theta = 1 - x^(2)`
`therefore sin theta = pm sqrt(1-x^(2))`
अब (2) से, `xsqrt(3) = cos((pi)/(2) -theta) = sin theta" "...(3)`
या `xsqrt(3) = pm sqrt(1-x^(2))" "` वर्ग करने पर, `3x^(2) = 1 -x^(2)`
या, `4x^(2) = 1 rArr x = pm (1)/(2)`
जाँच : जब `x = (1)/(2)`,
समीकरण (1) का `L.H.S. = cos^(-1) ((sqrt(3))/(2)) + cos^(-1) ((1)/(2))`
` = (pi)/(6) + (pi)/(3) = (pi)/(2) =` समीकरण (1) का R.H.S.
जब `x = - (1)/(2)`
समीकरण (1) का `L.H.S. = cos^(-1) (-(sqrt(3))/(2)) + cos^(-1) (-(1)/(2))`
`= pi - cos^(-1)((sqrt(3))/(2)) + pi - cos^(-1) ((1)/(2))`
` =pi - (pi)/(6) +pi-(pi)/(3) = (3pi)/(2) ne` समीकरण (1) R.H.S.
`therefore` केवल `x = (1)/(2)` ही हल है |