In Sec. 34, use expressions (13) and (14) to derive expressions (15) and (16) for |sin z|2 and |cos z|2.
(13) sin z = sin x cosh y + i cos x sinh y,
(14) cos z = cos x cosh y − i sin x sinh y,
where z = x + iy. To obtain expressions (13) and (14), we write
z1 = x and z2 = iy
in identities (5) and (6) and then refer to relations (12). Observe that once expression (13) is obtained, relation (14) also follows from the fact (Sec. 21) that if the derivative of a function
f (z) = u(x, y) + iv(x, y)
exists at a point z = (x, y), then
f'(z) = ux(x, y) + ivx(x, y).
Expressions (13) and (14) can be used (Exercise 7) to show that
(15) | sin z|2 = sin2 x + sinh2 y,
(16) | cos z|2 = cos2 x + sinh2 y.
Inasmuch as sinh y tends to infinity as y tends to infinity, it is clear from these two equations that sin z and cos z are not bounded on the complex plane, whereas the absolute values of sin x and cos x are less than or equal to unity for all values of x. (See the definition of a bounded function at the end of Sec. 18.)
A zero of a given function f (z) is a number z0 such that f(z0) = 0. Since sin z becomes the usual sine function in calculus when z is real, we know that the real numbers z = nπ (n = 0,± 1, ±2, . . .) are all zeros of sin z. To show that there are no other zeros, we assume that sin z = 0 and note how it follows from equation (15) that
sin2 x + sinh2 y = 0.
This sum of two squares reveals that
sin x = 0 and sinh y = 0.
Evidently, then, x = nπ (n = 0, ±1, ±2, . . .) and y = 0 ; that is,