Relatively simple question, that might not be simple to answer: I have noticed that there are ways of expressing every double angle formula of a given trigonometric function using only that function except for $\sin$ and $\csc$. That is,
$\sin2\theta=2\sin\theta\cos\theta=?$
$\cos2\theta=2\cos^2\theta-1$
$\tan2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}$
$\csc2\theta=\dfrac{1}{2}\csc\theta\sec\theta=?$
$\sec2\theta=\dfrac{\sec^2\theta}{2-\sec^2\theta}$
$\cot2\theta=\dfrac{\cot^2\theta-1}{2\cot\theta}$
For triple angle formulas, all 6 trig functions have expressions using only the given trig function. They are
$\sin3\theta=3\sin\theta-4\sin^3\theta$
$\cos3\theta=4\cos^{3}\theta-3\cos\theta$
$\tan3\theta=\dfrac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}$
$\csc3\theta=\dfrac{\csc^3\theta}{3\csc^2\theta-4}$
$\sec3\theta=\dfrac{\sec^3\theta}{4-3\sec^2\theta}$
$\cot3\theta=\dfrac{3\cot\theta-\cot^3\theta}{1-3\cot^2\theta}$
My question is: What are the the formulas, provided they exist, for $\sin2\theta$ and $\csc2\theta$ in terms of $\sin\theta$ and $\csc\theta$ respectively. If they do not exist, then some explanation as to why it is not possible would be most insightful.
Edit: I appreciate the answers so far, but what I really wanting to know is: Is there a known closed form (no piecewise-defined function) expression for the given expressions. Or if not, how to show that there is no such expression?