First slide

Trigonometric equations

Question

If $15 \sin^{4} x + 10 \cos^{4} x = 6$ , then $\tan^{2} x =$

Difficult

A

$\frac{1}{5}$
B

$\frac{2}{5}$
C

$\frac{2}{3}$
D

$\frac{1}{3}$

Solution

$W e h a v e 15 \sin^{4} x + 10 \cos^{4} x = 6 {(\sin^{2} x + \cos^{2} x)}^{2}$
$\Rightarrow 9 \sin^{4} x + 4 \cos^{2} x - 12 \sin^{2} x \cos^{2} x = 0$
$\Rightarrow {(3 \sin^{2} x - 2 \cos^{2} x)}^{2} = 0$
$\Rightarrow \tan^{2} x = 2 / 3$

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