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
  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: 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.

  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

    eix = cosx + isinx
    e-ix = cosx - isinx

Cheers,
Penny
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