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प्रश्न
Given `tan θ = 1/sqrt(5)`, what is the value of `(cosec^2θ - sec^2θ)/(cosec^2θ + sec^2θ)`?
योग
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उत्तर
Given: `tan θ = 1/sqrt(5)`
Step-wise calculation:
1. From `tan θ = "Opposite"/"Adjacent" = 1/sqrt(5)`
Take opposite = 1
Adjacent = `sqrt(5)`
Hypotenuse = `sqrt(1 + 5) = sqrt(6)`
2. So `sin θ = 1/sqrt(6), cos θ = sqrt(5)/sqrt(6)`.
Hence `cosec θ = sqrt(6)` and `sec θ = sqrt(6)/sqrt(5)`.
3. Compute squares: `cosec^2 θ = 6, sec^2 θ = 6/5`.
4. Form the expression: `(cosec^2 θ - sec^2 θ)/(cosec^2 θ + sec^2 θ)`
= `(6 - 6/5)/(6 + 6/5)`
= `(24/5)/(36/5)`
= `24/36`
= `2/3`
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