First slide

Trigonometric equations

Question

The sum of distinct solutions of the equation $(4 \tan x + \tan^{2} x + 1) = 2 \sqrt{2} \sin (x + \frac{π}{4}) (1 + \tan^{2} x)$ in the interval $[0, π]$ is:

Difficult

A

$\frac{π}{2}$
B

$\frac{4 π}{3}$
C

$2 π$
D

$\frac{7 π}{6}$

Solution

$\begin{matrix} (4 \tan x + \sec^{2} x) = 2 \sqrt{2} \sin (x + \frac{π}{4}) \sec^{2} x \\ \Rightarrow 4 \sin x \cos x + 1 = 2 (\sin x + \cos x) \end{matrix}$
$\Rightarrow 2 \sin x (2 \cos x - 1) - (2 \cos x - 1) = 0$
$\begin{matrix} \sin x = \frac{1}{2}, \cos x = \frac{1}{2} \\ \Rightarrow x = \frac{π}{6}, \frac{5 π}{6}, \frac{π}{3} \end{matrix}$

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