A person standing at the foot of a tower walks a distance of 3 meters from the tower and observes that the angle of elevation of the top of the tower is 30 o . He then walks a distance 4 meters perpendicular to the previous direction and observes the angle of elevation to be β . Then, cos ⁡ 2 β is equal to

Step-by-step solution

Let O be the foot of the tower O P O A = 3 m , A B = 4 m ∴ O B = 5 m   ∵ O A 2 + A B 2 = O B 2 ∴ In ∆ A O P , tan ⁡ 30 ° = h 3   ∴ h = 3 and in ∆ O P B , tan ⁡ β = h 5   ∴ h = 5 tan ⁡ β Thus, h = 3 = 5 tan ⁡ β ⇒ tan ⁡ β = 3 5 cos ⁡ 2 β = 1 - tan 2 ⁡ β 1 + tan 2 ⁡ β = 1 - 3 25 1 + 3 25 = 11 14