Question

Check whether the function f, defined by :
f (x)= cos 2x + tanx,
is periodic. If so, find its period. If f is not
periodic, define a functioning, such that
f — g is periodic.

Answer 1

Answer on Question #76394 – Math – Algebra

Question

Check whether the function $f$, defined by:

$f(x) = \cos 2x + \tan x$,

is periodic. If so, find its period. If $f$ is not periodic, define a functioning, such that

$f - g$ is periodic.

Solution

Let $x = 0 \Rightarrow \cos 0 + \tan 0 = 1 + 0 = 1$. We solve equation

$$\cos 2x + \tan x = 1; \quad \frac{1 - \tan^2 x}{1 + \tan^2 x} + \tan x = 1; \quad 1 - \tan^2 x + \tan x + \tan^3 x = 1 + \tan^2 x;$$ $$\tan x(\tan^2 x - 2\tan x + 1) = 0 \Rightarrow \tan x = 0, \tan x = 1$$ $$x = 0 \pm m\pi, \quad x = 45{}^\circ \pm m\pi, \quad m \text{ is integer}.$$

So, the period of our function is $\pi$. Check it.

$$f(x + \pi) = f(x)?$$ $$f(x + \pi) = \cos 2(x + \pi) + \tan(x + \pi) = \cos (2x + 2\pi) + \tan x = \cos 2x + \tan x = f(x).$$

Thus, the function $f$ defined by $f(x) = \cos 2x + \tan x$ is periodic. The period of the function is $\pi$.

Answer:

The function $f$ defined by $f(x) = \cos 2x + \tan x$ is periodic.

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