Hello, xMichaelx!

A Cone has a slant height and a semi-vertical angle

Prove that the radius of the inscribed sphere is given by:

. .

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The side view of the cone is

P is the center of the inscribed sphere: .

Note that

In right triangle

. . Hence: . .[1]

In right triangle .[2]

Equate [1] and [2]: .

Therefore: .