Question

If α+β=Φ and tanα=ktanβ then prove that sinα=k+1/k-1sin{α-β}

Answer 1

Let, α and β be the two parts of the angle θ.

Therefore, θ = α + β.

By question, θ = α - β. (assuming a >β)

and tan α/tan β = k 

⇒ sin α cos β/sin β cos α = k/1

⇒ (sin α cos β + cos α sin β)/(sin α cos β - cos α sin β) = (k + 1)/(k - 1), [by componendo and dividendo]

⇒ sin (α + β)/sin (α - β) = (k + 1)/(k - 1)

⇒ (k + 1) sin Ø = (k - 1) sin θ, [Since we know that α + β = θ; α + β = ф]

⇒ sin ф = (k - 1)/(k + 1) sin θ.