Question

prove that$sin 5x+sin3x/cos5x+cos3x=tan4x$

Answer 1

(Sin 5x + Sin 3 x) / ( Cos 5 x + Cos 3x )
  =  (2 Sin (5x+3x)/2 * Cos (5x-3x)/2 ) / 2 Cos (5x+3x)/2  *  Cos (5x-3x)/2
  =  Sin 4x Cos x / Cos 4x Cos x
 = tan 4x

Answer 2

(Sin 5x+Sin 3 x)
(Cos 5x+Cos 3x)

  =  2 Sin (5x+3x × Cos (5x-3x)
               2                    2     
     2 Cos (5x+3x)×Cos (5x-3x)
                2                 2

  =  Sin 4x Cos x
     Cos 4x Cos x

 = Sin 4x
    Cos 4x

 = tan 4x