I know that sine and cosine can be rewritten in terms of the real and complex parts of the exponential function as a result of Euler's formula.
My question is, can every trigonometric expression be written in terms of elementary trigonometric functions ($\sin$, $\cos$)? If not, why couldn't they be?
I would think that they could, although I understand that sometimes it may be prohibitive to do so since most trig identities can be derived from Euler's formula. Are there any cases when a trig expression absolutely cannot be written in terms of the elementary functions?
The only potential counterexamples I could think of would include some non trigonometric terms or factors.
I know that hyperbolic sine and cosine can be rewritten in terms of sine and cosine in the complex plane.