Question

If `tan θ = 1/2` then evaluate `((cos θ)/(sin θ) + (sin θ)/(1 + cos θ))`.

Answer 1

`tan θ = 1/2 = (BC)/(AB)`.

Let BC = x and AB = 2x.

∴ AC² = AB² + BC²

= (4x² + x²)

= 5x²

⇒ `AC = sqrt(5)x`.

∴ `sin θ = (BC)/(AC) = x/(sqrt(5)x) = 1/sqrt(5)` and `cos θ = (AB)/(AC) = (2x)/(sqrt(5)x) = 2/sqrt(5)`.

∴ `((cos θ)/(sin θ) + (sin θ)/(1 + cos θ)) = (cot θ + (sin θ)/(1 + cos θ))`

= `[2 + ((1/sqrt(5)))/(1 + 2/sqrt(5))]`

= `{2 + 1/((2 + sqrt(5))) xx ((2 - sqrt(5)))/((2 - sqrt(5)))}`

= `{2 - (2 - sqrt(5))}`

= `sqrt(5)`