Cosine of the angle of intersection of curve $y=(3^{x-1})\ln x$ and $y=(x^x-1)$ is

1

Point of intersection is (1, 0)

$\left(\frac{dy}{dx}\right)_I=3^{x-1}.\ln 3.\ln x+\frac{3^{x-1}}{x}$

$\left(\frac{dy}{dx}\right)_{II}=x^x(1+\ln x)$

Now, $\left(\frac{dy}{dx}\right)_I$ at (1, 0) is 1 and $\left(\frac{dy}{dx}\right)_{II}$ at (1, 0) is 1,

so angle of intersection is ‘0’ or they touch each other so cos θ = 1.