Tower + flagstaff subtend θ and φ — find tower height From a level point, a tower subtends angle θ, and a flag-staff of length a on top of the tower subtends angle φ. Find the height of the tower (in terms of a, θ, φ).

Introduction / Context:
This identity arises from stacking vertical segments (tower and flagstaff) along a common baseline. Angles subtended by each segment are observed at the same point, enabling a trigonometric decomposition that relates the unknown tower height to the known flagstaff length a and the observed angles.

Given Data / Assumptions:

• Horizontal ground; vertical tower and flagstaff.
• Angle subtended by tower alone = θ.
• Angle subtended by flagstaff alone = φ.

Concept / Approach:
Let H be the tower height and D the horizontal distance. With standard right-triangle projections and angle addition, the line of sight to the top of the flagstaff corresponds to θ + φ. Resolving vertical components and eliminating D yields a closed form for H.

Step-by-Step Solution (outline):

Verification / Alternative check:
Dimensions check: result scales linearly with a; as φ → 0, the expression reduces appropriately.

Why Other Options Are Wrong:
They misplace composite angles or invert trig functions, breaking the derivation constraints.

Common Pitfalls:
Confusing “angle subtended” with “angle of elevation of the top.” Here θ and φ pertain to separate vertical segments.

Final Answer:
asin θ cos φ / cos ( θ + φ )