A calculator is broken so that the only keys that still work are the $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, and $
\arctan$ buttons. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians, and you may assume all the values of $x$ below are positive. (a) Find, with proof, a sequence of buttons that will transform $x$ into $\frac{1}{x}$ . (b) Find, with proof, a sequence of buttons that will transform $\sqrt x$ into $\sqrt{x 1}$. (c) The display initially shows $1$. Prove that there is a sequence of buttons that will produce $\frac{3}{\sqrt{5}}$.
