Question

If `cot θ = (1)/sqrt(3)`, show that `((1 - cos^2θ)/(2 - sin^2θ)) = (3)/(5)`.

Answer 1

`cot θ = (1)/sqrt(3)`

⇒ `cot θ = (1)/(tan θ)` 

= `(1)/sqrt(3)`

= `"Base"/"Perpendicular"`

Hypotenuse = `sqrt(("Perpendicular")^2 + ("Base")^2`

= `sqrt((sqrt(3))^2 + 1`

= `sqrt(3 + 1)`

= 2

`cos θ = "Base"/"Hypotenuse"`

= `(1)/(2)`

`sin θ = "Perpendicular"/"Hypotenuse"` 

= `sqrt(3)/(2)`

To show: `(1 - cos^2θ)/(2 - sin^2θ) = (3)/(5)`

`(1 - cos^2θ)/(2 - sin^2θ) = (1 - (cosθ)^2)/(2 - (sinθ)^2)`

= `(1 - 1/4)/(2 - 3/4)`

= `(3/4)/(5/4)`

= `(3)/(5)`