CGPET 2016 Solved Paper

Given, cos‌θ=‌

cos2α+cos2β+cos2γ

sin‌2α+sin‌2β+sin‌2γ

......(i)
where, α,β,γ are the angles made by a line with positive direction of the axes of reference.
We know that, I2+m2+n2=1
∴ cos2α+cos2β+cos2γ=1.....(ii)[I=cos‌α,m=cos‌β‌and‌n=cos‌γ]
⇒ (1−sin2α)+(1−sin2β)+(1−sin2γ)=1
⇒ 3−(sin2α+sin2β+sin2γ)=1
⇒ sin2α+sin2β+sin2γ=2 ....(iii)
cos‌θ=

1

2

[from Eqs. (ii) and (iii)]
⇒ θ=cos−1(

1

2

)
⇒ θ=

π

3

=60°