Question

If `cot theta = 1/sqrt3` find the value of `(1 - cos^2 theta)/(2 - sin^2 theta)`

Answer 1

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`