Answer
$1$
Work Step by Step
Find $dy/d\theta$:
$\frac{dy}{d\theta}=\frac{d}{d\theta}(r\sin\theta)=\frac{d}{d\theta}(\cos 2\theta\sin\theta)=-2\sin2\theta\sin\theta+\cos2\theta\cos\theta$
Find $dx/d\theta$:
$\frac{dx}{d\theta}=\frac{d}{d\theta}(r\cos\theta)=\frac{d}{d\theta}(\cos2\theta\cos\theta)=-2\sin2\theta\cos\theta+\cos2\theta(-\sin\theta)=-2\sin2\theta\cos\theta-2\cos2\theta\sin\theta$
Find $dy/dx$:
$\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{-2\sin2\theta\sin\theta+\cos2\theta\cos\theta}{-2\sin2\theta\cos\theta-\cos2\theta\sin\theta}$
Find the slope of the tangent line at $\theta=\pi/4$:
$m=\frac{dy}{dx}=|_{\theta=\pi/4}=\frac{-2\sin(\pi/2)\sin(\pi/4)+\cos(\pi/2)\cos(\pi/4)}{-2\sin(\pi/2)\cos(\pi/4)-\cos(\pi/2)\sin(\pi/4)}=\frac{-2\cdot 1\cdot \sqrt{2}/2+0\cdot \sqrt{2}/2}{-2\cdot 1\cdot \sqrt{2}/2-0\cdot \sqrt{2}/2}=\frac{-\sqrt{2}}{-\sqrt{2}}=1$
Thus, the slope is $1$.
