Question

Given `tan θ = 1/sqrt(5)`, what is the value of `(cosec^2θ - sec^2θ)/(cosec^2θ + sec^2θ)`?

Answer 1

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`