Sum of modulus of complex numbers : $|\sin(z)|+|\cos(z)| \geq 1$

Question

I'm trying to establish if $|\sin(z)|+|\cos(z)|$ is greater than or equal to $1$.

I have tried to write out the expression in exponential form, but I don't really arrive at anything useful.

I would really appreciate any help! Thank you!

Answer 1

assume that $|\sin(z)|+|\cos(z)|<1$ squarinq we obtain $2|\sin(z)||\cos(z)|<0$ which is a contradiction.

Answer 2

Writing $w=e^{iz}$ and using the definition of $\sin z$ and $\cos z$, the inequality becomes equivalent to $$ |w^2-1|+|w^2+1|\ge2|w| $$ (for $w\ne0$, but it is trivial for $w=0$).

It is not restrictive to prove the squared inequality; we get $$ (w^2-1)(\bar{w}^2-1)+(w^2+1)(\bar{w}^2+1)+2|w^4-1|\ge4w\bar{w} $$ that reduces to $$ w^2\bar{w}^2+1+|w^4-1|\ge2w\bar{w} $$ (where all terms are real) so to $$ (w\bar{w}-1)^2+|w^4-1|\ge0 $$ which is true.