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.
Best 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.