 Solution

Where the original expression is defined, multiplication by $\cos x$ givesThus the graph repeats with period $\pi$ and has vertical asymptotesBetween consecutive asymptotes the extended cotangent branch decreases from $+\infty$ to $-\infty$, since $y^{\prime }=-1/(\cos x+\sin x{)}^2<0$. It passes through $(k\pi ,1)$ and $(\pi /4+k\pi ,1/2)$.

It is essential to preserve the domain when simplifying a function. At $x=\pi /2+k\pi$, the original tangent is undefined. The finite limiting value is zero, so the graph has holes at $(\pi /2+k\pi ,0)$, rather than actual zeros. The simplified expression extends across those points, but the original one does not.