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Trigonometrical Equations
medium
$\alpha=\sin 36^{\circ}$ is a root of which of the following equation
A
$10 x^{4}-10 x^{2}-5=0$
B
$16 x^{4}+20 x^{2}-5=0$
C
$16 x^{4}-20 x^{2}+5=0$
D
$16 x^{4}-10 x^{2}+5=0$
(JEE MAIN-2022)
Solution
$\cos 72^{\circ}=\frac{\sqrt{5}-1}{4}$
$\Rightarrow 1-2 \sin ^{2} 36^{\circ}=\frac{\sqrt{5}-1}{4}$
$\Rightarrow 4-8 \alpha^{2}=\sqrt{5}-1$4
$\Rightarrow 5-8 \alpha^{2}=\sqrt{5}$
$\Rightarrow\left(5-8 \alpha^{2}\right)^{2}=5$
$\Rightarrow 25+64 \alpha^{4}-80 \alpha^{2}=5$
$\Rightarrow 64 \alpha^{4}-80 \alpha^{2}+20=0$
$\Rightarrow 16 \alpha^{4}-20 \alpha^{2}+5=0$
Standard 11
Mathematics
