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Find the value of tan [1/2*cos^-1 (sqrt5...

Find the value of` tan [1/2*cos^-1 (sqrt5/(3))]`

Text Solution

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`tan[ 1/2 cos^-1(sqrt5/3)]`
`1/2 cos^-1(sqrt5/3) = theta`
`cos^-1(sqrt5/3) = 2 theta`
`cos 2theta = sqrt5/3`
`(1- tan^2 theta)/(1 + tan^2 theta) = sqrt5/3`
`3- 3 tan^2 theta = sqrt5 + sqrt5 tan^2 theta`
`sqrt((3 - sqrt5)/(3 + sqrt5))= tan theta`
`sqrt(((3- sqrt5)(3- sqrt5))/((3 + sqrt5)(3- sqrt5))) = tan theta`
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