First slide

Trigonometric equations

Question

The smallest positive $x$ satisfying the equation $\log_{\cos x} \sin x + \log_{\sin x} \cos x = 2$ is

Easy

A

$\frac{π}{4}$
B

$\frac{π}{6}$
C

$\frac{π}{2}$
D

$\frac{π}{3}$

Solution

We have $\log_{\cos x} \sin x + \log_{\sin x} \cos x = 2 = 1 + \frac{1}{1}$
$\Rightarrow \log_{\cos x} \sin x = 1$
$\Rightarrow \sin x = \cos x$
$\Rightarrow \tan x = 1$
$\Rightarrow x = \frac{π}{4}$

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