First slide

Trigonometric equations

Question

Number of solutions of the equation $4 \sin^{2} x + \tan^{2} x + \cot^{2} x + \cos e c^{2} x = 6$ in $[0, π]$ is

Moderate

A

0
B

2
C

4
D

8

Solution

$W e h a v e 4 \sin^{2} x + \tan^{2} x + \cot^{2} x + \cos e c^{2} x = 6$
$4 \frac{\frac{\sin^{2} x}{\cos^{2} x}}{\frac{1}{\cos^{2} x}} + \tan^{2} x + \cot^{2} x + 1 + c o t^{2} x = 6$
$\frac{4 \tan^{2} x}{s e c^{2} x} + \tan^{2} x + 2 c o t^{2} x = 5$
$L e t \tan x = t \Rightarrow \frac{4 t^{2}}{1 + t^{2}} + t^{2} + \frac{2}{t^{2}} = 5$
$\frac{4 t^{4} + t^{4} (1 + t^{2}) + 2 (1 + t^{2})}{t^{2} (1 + t^{2})} = 5$
$4 t^{4} + t^{4} + t^{6} + 2 + 2 t^{2} = 5 t^{2} + 5 t^{4}$
$\Rightarrow t^{6} - 3 t^{2} + 2 = 0 c l e a r l y + 1, - 1 a r e r o o t s$
using sythetic division
$\Rightarrow (t^{2} - 1) (t^{4} + t^{2} - 2) = 0$
$\Rightarrow t^{2} = 1, t^{2} = - 2 (n o t p o s s i b l e)$
$\Rightarrow t^{2} = 1$
$\Rightarrow t = \pm 1 \Rightarrow \tan x = \pm 1 ⇒ x = \frac{π}{4}, \frac{3 π}{4}$

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