Answer
The tangents are $y=-x$ and $y=x$.
Here is the graph.
Work Step by Step
Find the parameters $t$ corresponding to the point $(0,0)$:
$(x,y)=(0,0)$
$(\cos t,\sin t\cos t)=(0,0)$
$\cos t=0$ and $\sin t\cos t=0$
$\cos t=0$ and $2\sin t\cos t=0$
$\cos t=0$ and $\sin (2t)=0$
$t=\pm\frac{\pi}{2},\pm \frac{3\pi}{2},\ldots$ and $t=0,\pm\frac{\pi}{2},\pm \pi,\pm \frac{3\pi}{2},\pm 2\pi,\ldots$
$t=\pm\frac{\pi}{2},\pm \frac{3\pi}{2},\pm \frac{5\pi}{2},\ldots$
Find the slope of the tangent for $t$ above:
$m=\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{\frac{d}{dt}(\sin t\cos t)}{\frac{d}{dt}(\cos t)}=\frac{\cos t\cos t+\sin t(-\sin t)}{-\sin t}=\frac{\cos^2t-\sin^2 t}{-\sin t}=\frac{\cos 2t}{-\sin t}$
For $t=\frac{\pi}{2},\frac{3\pi}{2},\ldots$, $m=\frac{-1}{-1}=1$
For $t=-\frac{\pi}{2},-\frac{3\pi}{2},\ldots$, $m=\frac{-1}{1}=-1$
Then, $m=\pm 1$.
Find the equation of the tangent to the curve at $(0,0)$ with the slope $m=-1$:
$y-0=-1(x-0)$
$y=-1x$
$y=-x$
Find the equation of the tangent to the curve at $(0,0)$ with the slope $m=11$:
$y-0=1(x-0)$
$y=1x$
$y=x$
Thus, the tangents are $y=-x$ and $y=x$.
Here is the graph.
