Question

If `tan θ = 20/21` then prove that `((1 - sin θ + cos θ))/((1 + sin θ + cos θ)) = 3/7`.

Answer 1

Given: `tan θ = 20/21`

To prove: `((1 - sin θ + cos θ))/((1 + sin θ + cos θ)) = 3/7`

Proof:

`tan θ = 20/21 = (BC)/(AB)`.

Let BC = 20x and AB = 21x.

Then, AC² = (AB² + BC²)

= [(21x)² + (20x)²]

= (441x² + 400x²)

= 841x²

⇒ `AC = sqrt(841x^2)`

⇒ AC = 29x

∴ `sin θ = (BC)/(AC) = (20x)/(29x) = 20/29` and `cos θ = (AB)/(AC) = (21x)/(29x) = 21/29`.

∴ `((1 - sin θ + cos θ))/((1 + sin θ + cos θ)) = ((1 - 20/29 + 21/29))/((1 + 20/29 + 21/29))`

= `((30/29))/((70/29))`

= `3/7`