
Shortcut Trick
For two points on the same side of a tree, the height h is given by the formula:
h = d ÷ (cot θ1 − cot θ2), where d is the distance between the points.
Here, d = 4 m, θ1 = 45°, and θ2 = 60°.
h = 4 ÷ (cot 45° − cot 60°) = 4 ÷ (1 − 1/√3) = 4√3 ÷ (√3 − 1).
Rationalizing the denominator gives h = 2√3(√3 + 1) m.
∴ The correct answer is 2√3(√3 + 1) m.

Alternate Method
Given:
Distance between two points (CD) = 4 m
Angle of depression to point C (θ1) = 45°
Angle of depression to point D (θ2) = 60°
Formula Used:
tan θ = Perpendicular / Base

Calculations:
Let AB = h be the height of the tree and BD = x be the distance from the base to point D.
In right-angled triangle ABD:
⇒ tan 60° = AB / BD
⇒ √3 = h / x
⇒ x = h / √3 ---(i)
In right-angled triangle ABC:
⇒ tan 45° = AB / BC
⇒ 1 = h / (x + 4)
⇒ x + 4 = h
⇒ x = h − 4 ---(ii)
Equating (i) and (ii):
⇒ h / √3 = h − 4
⇒ h = h√3 − 4√3
⇒ 4√3 = h(√3 − 1)
⇒ h = 4√3 / (√3 − 1)
Multiplying by conjugate (√3 + 1):
⇒ h = [4√3 × (√3 + 1)] / [(√3 − 1) × (√3 + 1)]
⇒ h = 4√3(√3 + 1) / (3 − 1)
⇒ h = 4√3(√3 + 1) / 2
⇒ h = 2√3(√3 + 1)
∴ The correct answer is 2√3(√3 + 1) m.

Additional Information
Angle of Elevation and Depression
The Angle of Elevation is measured upwards from the horizontal, while the Angle of Depression is measured downwards. They are numerically equal as alternate interior angles.
Points on Opposite Sides
If points are on opposite sides of the tree, the distance between them is d = h(cot θ1 + cot θ2).
Trigonometric Ratios
Crucial values: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.