Prove that the evolute of an astroid is another astroid using formulas. Prove that the evolute of an astroid is another astroid using formulas.
 A: You've made a mistake in the final calculation of the $x$-coordinate of the evolute. (Note: In what follows, I'm suppressing explicit dependence on $t$ where it's unambiguous, writing $x$ for $x(t)$, etc.)
$$
(x')^2 + (y')^2 = 9 \sin^2 t \cos^2 t 
$$
and
$$
x' y'' - x'' y' = -9 \sin^2 t \cos^2 t,  
$$
the coordinates of the evolute simplify to
$$
\left\{
\begin{align} 
a &= x + y' \\
b &= y - x' 
\end{align}
\right.
$$
After substituting explicit expressions for $t$, we get
$$
\left\{
\begin{align} 
a &= 3\cos t - 2\cos^3 t \\
b &= 3\sin t - 2\sin^2 t 
\end{align}
\right.
$$
Why is this also the equation of an astroid? The graph seems to show that the evolute is a copy of the original astroid rotated by $\frac{\pi}{4}$ and scaled by a factor of $2$. In other words, using the standard formulas for rotation and noting that
$$
\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}, 
$$
we would like to show that
$$
\left\{
\begin{align} 
u &= 2 \cdot \frac{x - y}{\sqrt{2}} \\
v &= 2 \cdot \frac{x + y}{\sqrt{2}}
\end{align}
\right.
$$
parametrizes the evolute.
However, this doesn't work, as you can check by substituting in the formulas for $x$ and $y$ and noting that the resulting $(u, v)$ does not equal the parametrization $(a, b)$ calculated before.
The problem is that in the rotated coordinates, we need to shift the parameter, so that each $t$ corresponds to the same point on the evolute astroid as the original one. For example, at $t=0$, we have the point $(x, y) = (1, 0)$ right in the middle of one of the curving sides of the original astroid, but the corresponding point on the evolute (center of the osculating circle) comes out to the cusp $(a, b) = (\sqrt{2}, \sqrt{2})$. If we intend to recover this evolute curve via transformations of the original astroid, we need to $t=0$ to correspond to the cusp.
Hence, the formulas that we should verify look like:
$$
\left\{
\begin{align} 
a(t) &= 2 \cdot 
\frac{x\bigr(t-\frac\pi4\bigr) - y\bigr(t-\frac\pi4\bigr)}{\sqrt{2}} \\
b(t) &= 2 \cdot 
\frac{x\bigr(t-\frac\pi4\bigr) + y\bigr(t-\frac\pi4\bigr)}{\sqrt{2}}
\end{align}
\right.
$$
You should expand each of these expressions and check that they indeed recover the formulas for $a$ and $b$ above!
