Answer 1

Given `tan theta = 1/sqrt2`

We have to find the value of the expression `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + cot^2 theta)`

We know that,

`1 +cot^2 theta = cosec^2 theta`

`=> cosec^2 theta - cot^2 theta = 1`

Therefore, the given expression can be written as

`(cosec^2 theta - sec^2 theta)/(cosec^2 theta + cot^2 theta) = (cosec^2 theta - sec^2 theta)/(1 + cot^2 theta + cot^2 theta)`

`tan theta = 1/sqrt2 => cot theta = sqrt2`

`(cosec^2 theta - sec^2 theta)/(1 + 2 cot^2 theta) = (1 + cot^2 theta -  (1 + tan^2 theta))/(1 + 2 cot62 theta)`             (since `sec^2 theta    =1 + tan^2 theta`)

`= (cot^2 theta - tan^2 theta)/(1 + 2 cot^ theta)`

`= ((sqrt2)^2 - (1/sqrt2)^2)/(1 + 2 xx (sqrt2)^2)`

`= 3/10`

Hence, the value of the given expression is 3/10