different forms of complex sine function

Question

Pick out the true statements:
(a) $|\sin z|≤1 ∀z∈\mathbb{C}$.
(b) $\sin^2z+\cos^2z=1 ∀z∈\mathbb{C}$.
(c) $\sin z =(e^{iz}-e^{-iz})/2 ∀z∈\mathbb{C}$.

(a) is not true for large z.
(b) true.
(c) true
Am I correct?

Answer 1

(a) is not true but I don't know what you mean "large" $z$? I suppose you mean large in modulus? You can always use Liouville's Theorem of course for a more general result (or even the little Picard theorem for stronger results).

(c) is not true as you need a $i$ in the denominator.

(b) is true but you might want to prove it.

Answer 2

sin z cannot be bounded take z_n=in, then sin z_n=[Exp(-n)-Exp(n)]/2 this diverges to -infinity as n increases