Collapsing a Linear Combination

For a specific constant $\phi$, the identity below holds for all values of $\theta$:$$4\cos\theta-3\sin\theta=5\cos(\theta+\phi)$$If $\phi\in \left[0,\frac{\pi}{2}\right)$, find the value of $\sin\phi +\cos\phi$. The answer is in the form $\frac{\alpha}{\beta}$ for coprime integers $\alpha$ and $\beta$. Input the answer as the sum $\alpha +\beta$.