Derivation through first principle sin 2x?

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Explanation:

Consider the attached diagram.

Arc RB is centered at O and has a radius of 1. Angle ROD is designated angle α. It is duplicated as angle DOB, meaning that OD is the perpendicular bisector of chord RB. Triangles DFB and RGD are congruent (by HA). They are similar to triangle RDO by AA similarity.

Now, we can use the definitions of sine and cosine to find some segment lengths. Since the circle is of radius 1, the length RD is sin(α). Then, in triangle RGD, the length DG is sin(α)·cos(α). As we said, BF is congruent to DG, so the length BH is 2·sin(α)·cos(α). Again, by the definition of sine, BH = sin(2α). So, we have ...

  sin(2α) = 2·sin(α)·cos(α) . . . . the desired relationship

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step-by-step explanation: