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Prove that cos ^(-1) x = 2 sin ^(-1).sq...

Prove that `cos ^(-1) x = 2 sin ^(-1).sqrt(1-x)/(2)`

Text Solution

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Let `cos^(-1) x = theta`
`rArr" "x - cos theta`
R.H.S `= 2 sin sqrt((1-x)/(2)) = 2sin. sqrt((1-costheta))/(2)`
`= 2 sin ^(-1).sqrt((2 sin^(2)((theta)/(2)))/(2)) = 2 sin (sin.(theta)/(2)`)
` 2.(theta)/(2)= theta= cos^(-1) x= L.H.S`
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