trigonometry - Can all trigonometric expressions be written in terms of sine and cosine?

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.

Answer

$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$

$$\cot\theta=\frac{\cos\theta}{\sin\theta}$$

$$\sec\theta=\frac{1}{\cos\theta}$$

$$\csc\theta=\frac{1}{\sin\theta}$$

and since in the complex plane, we have

$$\begin{align}
\cosh\theta&=\phantom{-i}\cos{i\theta} \\
\sinh\theta&=-i\sin{i\theta} \\

\tanh\theta&=-i\tan{i\theta} \\
\coth\theta&=\phantom{-}i\cot{i\theta} \\
\operatorname{sech}\theta&=\phantom{-i}\sec{i\theta} \\
\operatorname{csch}\theta&=\phantom{-}i\csc{i\theta}
\end{align}$$

And thus you can continue the definitions of the hyperbolic functions by substituting the sine and cosine definitions of the the remaining 4 trigonometric functions.