Question

If alpha+ beta = gamma,then cos^2beta+cos^2 alpha-2cos alpha.cos beta.cos gamma=​

Answer 1

Answer:

Heya mate your answer

Step-by-step explanation:

We have to prove,

cos2α+cos2β+cos2γ=1+2cosαcosβcosγ

We can write it as

cos2α+cos2β+cos2γ−1=2cosαcosβcosγ

L.H.S.=cos2α+cos2β+cos2γ−1

=cos2α+cos2β−1(1−cos2γ)

=cos2α+cos2β−sin2γ

=cos2α+cos2β−sin2(α+β) (As γ=α+β)

=cos2α+cos2β−(sinαcosβ+cosαsinβ)2

=cos2α+cos2β−sin2αcos2β−cos2αsin2β−2sinαcosβsinβcosα

=cos2α(1−sin2β)+cos2β(1−sin2α)−2sinαcosβsinβcosα

=cos2αcos2β+cos2βcos2α−2sinαcosβsinβcosα

=2cos2αcos2β−2sinαcosβsinβcosα

=2cosαcosβ(cosαcosβ−sinαsinβ)

=2cosαcosβcos(α+β)

=2cosαcosβcosγ=R.H.S