Why is radians the only unit for which I can use take $\sin(\sin(x))$ (Why is it the the default unit of trigonometry). This does not work if change the definition of a radian. But works if we change the definition of $\pi$ itself
Is this due to how we derive the Taylor Polynomial of $\sin(x)$ and $\cos(x)$ ? or How the differential of $\sin x$ in radians is $\cos x$ ?(i.e because $\frac{\sin x}{x}$ converges to $1$ as $x\to 0$)
Or is it because of some convention or assumption somewhere?
$\sin(\sin x)$ could be evaluated if $x$ is in radian or in degrees.
You will get different answers so we need to clarify which unit is used.
For example in radian $$\sin(\sin(90)) \approx 0.7795$$ but in degrees $$\sin(\sin(90))\approx0.017$$
The trig formulas for differentiation or integration are only valid in radians, but the trig identities are valid in any unit.
For instance $$ \sin^2 (x) + \cos^2(x) =1$$ is valid in any unit as well as $$\tan (x) = \frac {\sin (x)}{\cos(x)}$$