First slide

Trigonometric equations

Question

$\log_{\tan x} (2 + 4 \cos^{2} x) = 2$ , then $x =$

Easy

A

$\frac{π}{4}$
B

$\frac{π}{6}$
C

$\frac{- π}{4}$
D

$\frac{π}{3}$

Solution

We have $\log_{\tan x} (2 + 4 \cos^{2} x) = 2$
$\Rightarrow 2 + 4 \cos^{2} x = \tan^{2} x$
$\Rightarrow 3 + 4 \cos^{2} x = \sec^{2} x$
$\Rightarrow 4 \cos^{4} x + 3 \cos^{2} x - 1 = 0$
$\Rightarrow (4 \cos^{2} x - 1) (\cos^{2} x + 1) = 0$
$\Rightarrow \cos x = \pm \frac{1}{2}$

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