How to prove $|\sin(a)|+|\cos(a+b)|-|\cos(b)|\geq0$

Question

Prove $$|\sin(a)|+|\cos(a+b)|-|\cos(b)|\geq0$$ I want to use this as a lemma, but I dont know how to prove it.

Answer 1

We have $$\eqalign{|\cos(a-c)| &=|\cos a\cos c+\sin a\sin c|\cr &\le|\cos a|\,|\cos c|+|\sin a|\,|\sin c|\cr &\le|\cos c|+|\sin a|\ .\cr}$$ Now substitute $c=a+b$.