I.8.2 Show that cos 22 = cos + sinhy, where =+iy. Find all zeros and periods of cos z. Solution. Use trigonometric formulas from page 29 and 30 in CA we have, cos = cos(x + y) = cos r cos (iy)-sin r sin (iy) = cos r cosh yi sin x sinh y. Now take the modulus squared, and use cosh y = 1 + sinh y, |cos z = cos x cosh y + sin x sinh y = = cosr(1+sinhy) + sin z sinh y = = cos r+(cosr+sinr) sinhy=cosr+sinh y. The identity for cos 22 shows that the only zeros of cos z are the zeros of cos on the real axis, because cos x = 0 cos z=0 =+m, m = 0, 1, 2..... Translation by any period A of cos z sends zeros to zeros. Thus any period is an integral multiple of x, and since odd integral multiples are not periods. the only periods of cos z are 2n. -