Why do I get: $\cos((\frac1x)^4)\sin(x^6) = 0?$

Question

Can someone please help me understand what is wrong in the following?

$$\cos(\frac1{x^4})\sin(x^6) = \frac14\frac1i[(e^\frac{i}{x^4} + e^\frac{-i}{x^4})(e^{ix^6} - e^{-ix^6})] = \frac14\frac1i[e^{-x^2} - e^{x^2} + e^{x^2} - e^{-x^2}] = 0$$

Entering $cos(\frac1{x^4})*sin(x^6)$ into WolframAlpha gives me an actual function, which makes sense because I understand plugging in $x=1$ (for example) into $cos(\frac1{x^4})*sin(x^6)$ doesn't give $0$, so what is going on?

Answer 1

It appears you have computed for example

$$e^{i/x^4} \cdot e^{ix^6} = e^{-x^2}$$

using the false "rule" $e^a e^b = e^{ab}$. The true rule is $e^a e^b = e^{a+b}$, so

$$ e^{i/x^4} \cdot e^{i x^6} = e^{i/x^4 + ix^6} $$

and the result does not simplify very much after the four terms are multiplied out.