Period of the function cos (2tanx)+cos (2cotx)

Question

The answer is given as pi/2 . Is it because this function is undefined after every interval of pi/2 or is there some other explanation.

Answer 1

$$\tan\left(x+\frac\pi2\right)=-\cot x$$ and $$\cot\left(x+\frac\pi2\right)=-\tan x.$$

As the cosine is an even function, the minus signs are absorbed and the period is at most $\dfrac\pi2$, as

$$f\left(x+\frac\pi2\right)=f(x).$$

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We can confirm that this is the smallest period from the fact that the function has an unbounded slope only at the multiples of $\dfrac\pi2$.