Trigonometric inequality $|\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge1$ for all real $a_i$

Question

Prove that for all real numbers $a_1,a_2,...,a_n$ the following inequality holds: $$ |\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge 1 $$

Answer 1

HINT: $\sin^2x+\cos^2x=1$, $\sqrt{a^2+b^2}\leq|a|+|b|$, $|c+d|\leq|c|+|d|$.

Answer 2

HINT: $$\sin(a_1+a_2)=\sin(a_1)\cos(a_2)+\sin(a_2)\cos(a_1),$$ so $$ |\sin(a_1+a_2)|\le|\sin(a_1)|+|\sin(a_2)|.$$ Now you can use induction.