calculus

9. Recall that there is a number i with the properties i=i.i=-1, P=i.i=-1.i= -i, it = 12 .12 = -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. (a) Substitute ir into the Maclaurin series for cosh(r) and verify that cosh(ix) = cos(r). (b) Substitute ix into the Maclaurin series for sinh(x) and verify that sinh(ix) = i sin(). (c) Use the result e" = cosh(x) + sinh(x) to prove Euler's Formula ez = cos(x) + isin(x) (d) Use the result of part (c) to show that el + 1 = 0