Çözüldü Trigonometry - Euclid's Formula for The Right Triangle

Given a right triangle ABC with the right angle at the vertex A; in terms of the angle ABC, what is the height |AH| to the hypotenuse which is four units long?
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∡ABC = α
|BH| = y
|CH| = 4 - y
Based on Pythagoras' Theorem and using Euclid's right triangle properties;
|AH|^2 = |BH|·|CH| = y(4 - y) ]....(I)
tanα = |AH| / y ⇒ y = |AH|·cotα....(II)
Substituting the equation (II) to (I), |AH|^2 = |AH|·(cotα)[ 4 - |AH|·(cotα) ] and simplifying;
|AH| = 4cotα - |AH|·[ (cotα)^2 ], organizing to leave |AH| alone on the left side;
|AH| = 4cotα / [ 1 + (cotα)^2 ], switching to sine and cosine functions;
|AH| = (4cosα / sinα) / [ 1 / (sinα)^2 ] and after the final simplification;
|AH| = 4·sinα·cosα, using the trigonometric identity sin2θ = 2·sinθ·cosθ
|AH| = 2sin2α