
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: e^{ix} = 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 cos^{2}x + sin^{2}x = 1 and tan^{2}x + 1 = sec^{2}x
 check that your definitions are consistent with cos2x = cos^{2}x  sin^{2}x 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: e^{x} = 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
e^{ix} = cosx + isinx
e^{ix} = cosx  isinx
Cheers,
Penny
Go to Math Central

