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:

1. substitute -x for x to find e^-ix, simplifying your answer
   
   
2. use the given and part a to find an identity for cosx making no reference to trig functions
   
   
3. find an identity for sinx
   
   
4. find an identity for tanx. Then put it in a form where you are not "stacking fractions."
   
   
5. use your new "definitions" to confirm that cos²x + sin²x = 1 and tan²x + 1 = sec²x
   
   
6. check that your definitions are consistent with cos2x = cos²x - sin²x and two other identities of your choice.
   
   
7. 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

Here is a start.

1. substitute -x for x to find e^-ix, simplifying your answer
   
   e^-ix = cos(-x) + isin(-x) = cosx - isinx
   
2. 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