Incorrect calculations: tan(x) and cot(x) near x=k*π

Exploring the computational errors of the tangent function tan(x) and cotangent function cot(x) in MATLAB near numbers around $k\,\pi (k\neq0).$

Example 1. Suppose $x=65.973446\approx21\,\pi.$ Calculate $\tan(x)$ and $\cot(x)$ in MATLAB.

Let's just post the figure directly.

From the above figure, it can be seen that the two outputs from MATLAB are
$2.746143\red{375848801}e-07$ and $3.6414704\red{66526212}e+06.$
However, the correct values with 16 significant digits are 0.2746143419922914e-6 and 0.3641470408082585e7, respectively (as provided by ISRealsoft).
Thus, the outputs of MATLAB contain 9 and 8 incorrect digits, with error rates of 9/16 = 56.25% and 8/16 = 50% respectively for the significant figures.

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