A line makes an angle θ with the x- axis and the y- axis. If it makes an angle α with the z- axis such that sin 2 ⁡ α = 3 sin 2 ⁡ θ, then cos ⁡ 2 θ is equal to

Step-by-step solution

Direction cosines of the line are cos ⁡ θ , cos ⁡ θ , cos ⁡ α As we know that cos 2 ⁡ θ + cos 2 ⁡ θ + cos 2 ⁡ α = 1 ⇒ 2 cos 2 ⁡ θ = sin 2 ⁡ α … … i Also, sin 2 ⁡ α = 3 sin 2 ⁡ θ (Given) ⇒ 2 cos 2 ⁡ θ = 3 sin 2 ⁡ θ ⇒ tan 2 ⁡ θ = 2 3 cos ⁡ 2 θ = 1 - tan 2 ⁡ θ 1 + tan 2 ⁡ θ = 1 - 2 3 1 + 2 3 = 1 5