Question

Chứng minh: sin(x+a)+sin(x+2a)+sin(x+3a)+...+sin(x+100x)=0

Answer 1

\(A=cosx+cos\left(n+y\right)+cos\left(x+2y\right)+...+cos\left(x+ny\right)=\left(n+1\right)cosn\)

\(\dfrac{sinx+sin3x+sin5x+...+sin\left(2n-1\right)x}{cosx+cos3x+cos5x+...+cos\left(2n-1\right)x}\) 

                                            \(=tan\left(nx\right)\)

\(sinx+sin2x+sin3x+...+sinnx\)

               \(=\dfrac{sin\dfrac{nx}{2}sin\dfrac{\left(n+1\right)x}{2}}{sin\dfrac{x}{2}}\)

\(cosx+cos2x+cos3x+cosnx\)

              \(=\dfrac{sin\dfrac{nx}{2}cos\dfrac{\left(n+1\right)x}{2}}{sin\dfrac{x}{2}}\)