Prove Sine of sum equals Sum of Sines by Induction

Use mathematical induction to show that there exists real numbers $a_1+a_2+a_3+cdots+a_n$ such that $|a_i|le 1$ for $i=1, 2, 3, …, n$ and such that $sin (x_1+x_2+x_3+cdots+x_n)= a_1sin x_1 + a_2sin x_2 + a_3sin x_3 + cdots+a_nsin x_n$.