First slide

Trigonometric equations

Question

The sum of all solutions in $[0, 4 π]$ of the equation $\tan + \cot x + 1 = \cos (x + \frac{π}{4})$ is

Easy

A

$3 π$
B

$\frac{π}{2}$
C

$\frac{7 π}{2}$
D

$4 π$

Solution

$W e h a v e \tan x + co t x = \cos (x + \frac{π}{4}) - 1$
$\tan x + \cot x \leq - 2$ and $\cos (x + \frac{π}{4}) - 1 \geq - 2$
It implies that equality holds when both are $- 2$
$\cos (x + \frac{π}{4}) = - 1 \Rightarrow x + \frac{π}{4} = (2 n + 1) π, n \in z$
$\Rightarrow x = \frac{3 π}{4}$ or $\frac{11 π}{4} \Rightarrow s u m = \frac{7 π}{2}$

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