Question

In any triangle ABC ,prove that
a cos A + b cos B + c cos C = 2a sin B sin C

Answer 1

Let (C)(C) be a unit circle, and M∈(C)M∈(C) . Also, we will denote ∠IOM∠IOM as θθ (see the diagram). From the unit circle definition, the coordinates of the point MM are (cosθ,sinθ)(cos⁡θ,sin⁡θ) . And so, OC¯¯¯¯¯¯¯¯OC¯ is cosθcos⁡θ and OS¯¯¯¯¯¯¯OS¯ is sinθsin⁡θ . Therefore, OM=OC¯¯¯¯¯¯¯¯2+OS¯¯¯¯¯¯¯2−−−−−−−−−√=cos2θ+sin2θ−−−−−−−−−−−√OM=OC¯2+OS¯2=cos2⁡θ+sin2⁡θ . Since MM lies in the unit circle, OMOM is the radius of that circle, and by definition, this radius is equal to 11 . It immediately follows that: cos2θ+sin2θ=1cos2⁡θ+sin2⁡θ=1

Answer 2

hope it will provide you better explanation.