Hi. I'm a junior in high school and this is a challenge problem that I was assigned for my Analysis class. We are allowed to use absolutely any sources to solve and understand it. Here it is: Given: eix = cosx + isinx substitute -x for x to find e-ix, simplifying your answer use the given and part a to find an identity for cosx making no reference to trig functions find an identity for sinx find an identity for tanx. Then put it in a form where you are not "stacking fractions." use your new "definitions" to confirm that cos2x + sin2x = 1 and tan2x + 1 = sec2x check that your definitions are consistent with cos2x = cos2x - sin2x and two other identities of your choice. now repeat a through e for the hyperbolic cos (coshx) and the hyperbolic sin (sinhx) defined as follows: ex = coshx + sinhx where coshx is an even function and sinhx is an odd function. (By the way, tanhx = (sinhx)/(coshx) and sechx = 1/(coshx). Your identities in part e will not be identical to those for the equivalent trig functions. Thank you very much for your time. I look forward to hearing from you. Thanks again, Peter Hi Peter, Here is a start. substitute -x for x to find e-ix, simplifying your answer e-ix = cos(-x) + isin(-x) = cosx - isinx use the given and part a to find an identity for cosx making no reference to trig functions Add the equations eix = cosx + isinx e-ix = cosx - isinx Cheers, Penny Go to Math Central