Matematické Fórum

Skrytý text:
First we denote



and using them we express :



It is obvious that the third identity implies that .

Now we express in the trigonometric form:



From the second identity and Moivre's Theorem we deduce that



It means that . Moreover, and a real number on the right-hand side imply that



The sine and cosine function at can be expressed in the following way:



Now we use the binomial theorem and modify the first identity:



We know that , see the second identity. Hence



The last equation is satisfied if and only if



With respect to the fact that the cosine function is bounded from above by 1 we get the only solution:



The second equation is fulfiled immediately.

Hence the problem has just one solution: .