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प्रश्न
if `tan theta = 1/sqrt2` find the value of `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + cot^2 theta)`
उत्तर
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
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