8. The hyperbolic cosine and hyperbolic sine functions are defined by the following equa- tions cosh(x) = ? (e" +e-") sinh(x) = 5(ex - e-z). [1] (a) Prove that -(cosh(x)) = sinh(x) and ( (sinh(x)) = cosh(x) [1] (b) Prove the identity e" = cosh(x) + sinh(x). [8] (c) Show that 22k+1 cosh(a) = * = 0 (2k)! and sinh(x) = > K = 0 (2k + 1)! and that each of these Maclaurin series has interval of convergence I = (-00, 00). 9. Recall that there is a number i with the properties 1 =i.i= -1, 83=12.i=-1 .i= -i, it=12.1? =-1. -1=1 This number even obeys all the usual laws of arithmetic; for example, i . 1 = i, i . 0 = 0, and i(a + b) = ia + ib. [3] (a) Substitute ix into the Maclaurin series for cosh(r) and verify that cosh(ix) = cos(x). [3] (b) Substitute ix into the Maclaurin series for sinh(x) and verify that sinh(ix) = i sin(x). [2] (c) Use the result e" = cosh(x) + sinh(x) to prove Euler's Formula ell = cos(x) + i sin(x) [1] (d) Use the result of part (c) to show that el + 1 = 0