Question

Sa se arate ca sinx+ sin3x +sin5x =(1 +2cos2x) sin3x , ∀X∈R.

Answer 1

Avem formula
$\sin{a}+\sin{b}=2\sin{\frac{a+b}{2}}\cos{\frac{a-b}{2}}$
Daca luam in considerare ultimul si primul termen al sumei
$\sin{5x}+\sin{x}=2\sin{\frac{5x+x}{2}}\cos{\frac{5x-x}{2}}=2\sin{3x}\cos{2x}$
Inlocuim aceasta suma in suma initiala
$\sin{3x}+2\sin{3x}\cos{2x}=(1+2\cos{2x})\sin{3x}$