If tan (alpha /2):tan (beta/2)=1:square root of 3,show that cos beta = 2 cos alpha - 1 / 2- cos - Maths - Trigonometric Functions

Question

If tan (alpha /2):tan (beta/2)=1:square root of 3,show that cos
beta = 2 cos alpha - 1 / 2- cos alpha

Rajat Sethi · Accepted Answer

tanα2:tanβ2=1:3⇒1-cosα1+cosα1-cosβ1+cosβ=13
[ Using
tan(A2)=1-cosA1+cosA]⇒3×(1-cosα1+cosα)=(1-cosβ1+cosβ)
[ Squaring both
sides]⇒3(1+cosβ-cosα-cosαcosβ)=1-cosβ+cosα-cosαcosβ⇒2+4cosβ-4cosα-2cosαcosβ=0⇒2cosβ(2-cosα)=4cosα-2⇒cosβ(2-cosα)=2cosα-1⇒cosβ=2cosα-12-cosα

If tan (alpha /2):tan (beta/2)=1:square root of 3,show that cos beta = 2 cos alpha - 1 / 2- cos alpha