Two successive elevations after walking a: From a point A, the angle of elevation of the top of a vertical tower is α. After walking a metres straight toward the tower, the angle becomes β (β > α). Express the height of the tower in terms of a, α, β.

Introduction / Context:
This is the standard two-elevation formula for a vertical tower when an observer moves a known distance a toward the tower, changing the angle from α to β. A sine-rule derivation yields a compact expression for the height.

Given Data / Assumptions:

• Initial elevation = α; after walking a toward tower, elevation = β.
• Height of tower = H.
• Both observations on the same straight line toward the tower.

Concept / Approach:
Let initial horizontal distance be x. Then tan α = H / x and tan β = H / (x − a). Eliminating x gives H = a * tan α * tan β / (tan β − tan α). Using the sine addition identity, this equals a * sin α * sin β / sin(β − α).

Step-by-Step Solution:

Verification / Alternative check:
Plug small numeric values (e.g., α = 30°, β = 60°, a = known) to confirm both tan-form and sine-form give identical H.

Why Other Options Are Wrong:
They have wrong sign in denominator, extra factors, or inverted ratios; only option (a) matches the proven identity.

Common Pitfalls:
Using degrees with calculator in radian mode; mixing tan-based and sin-based expressions mid-derivation; forgetting that β > α so denominator is positive.

Final Answer:
a Sin α Sin β/ Sin(β - α)