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Question
if `cot theta = 1/sqrt3` find the value of `(1 - cos^2 theta)/(2 - sin^2 theta)`
Solution
Given `cot theta = 1/sqrt3`
We have to find the value of the expression `(1 - cos^2 theta)/(2 - sin^2 theta)`
We know that
`1 + cot^2 theta = cosec^2 theta`
`=> cosec^2 theta = 1 + (1/sqrt3)^2 `
`=> cosec^2 theta = 4/3`
Using the identity `sin^2 theta + cos^2 theta =1` we have
`(1 - cos^2 theta)/(2 - sin^2 theta) = (sin^2 theta)/(2 - sin^2 theta)`
`= (1/(cosec^2 theta))/(2 - 1/(cosec^2 theta))`
`= 1/(2 cosec^2 theta - 1)`
`= 1/(2 xx 4/3 - 1)`
`=3/5`
Hence, the value of the given expression is 3/5
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