Incorrect calculations: tan(x) and cot(x) near x=k*π+π/2

Exploring the computational errors of the tangent function tan(x) and cotangent function cot(x) in MATLAB near numbers around $k\,\pi+\frac{\pi}{2}.$

Example 1. Suppose $x=39.26990817\approx12\,\pi+\frac{\pi}{2}.$ Calculate $\tan(x)$ and $\cot(x)$ in MATLAB.

Let's just post the picture directly.

However, the correct values with 16 significant digits are -0.7837941516239115e10 and -0.1275845192169577e-9, respectively (as provided by ISRealsoft).
Thus, the output of MATLAB contains only 4 and 5 correct digits, with error rates of (16-4)/16 = 75% and (16-5)/16 = 68.75% respectively for the significant figures.

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