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Trigonometrical Equations
hard
જો $e ^{\left(\cos ^{2} x+\cos ^{4} x+\cos ^{6} x+\ldots \ldots \infty\right) \log _{e} 2}$ એ સમીકરણ $t ^{2}-9 t +8=0,$ નું સમાધાન કરે, તો $\frac{2 \sin x}{\sin x+\sqrt{3} \cos x}\left(0 < x < \frac{\pi}{2}\right)$ નું મૂલ્ય .......... થાય.
A
$2 \sqrt{3}$
B
$\frac{3}{2}$
C
$\sqrt{3}$
D
$\frac{1}{2}$
(JEE MAIN-2021)
Solution
$e ^{\left(\cos ^{2} \theta+\cos ^{4} \theta+\ldots . . \infty\right) \ell n ^{2}}=2^{\cos ^{2} \theta+\cos ^{4} \theta+\ldots \infty}$
$=2^{\cot ^{2} \theta}$
Now $t^{2}-9 t+9=0 \Rightarrow t=1,8$
$\Rightarrow \quad 2^{\cot ^{2} \theta}=1,8 \Rightarrow \cot ^{2} \theta=0,3$
$0<\theta<\frac{\pi}{2} \Rightarrow \cot \theta=\sqrt{3}$
$\Rightarrow \quad \frac{2 \sin \theta}{\sin \theta+\sqrt{3} \sin \theta}=\frac{2}{1+\sqrt{3} \cot \theta}=\frac{2}{4}=\frac{1}{2}$
Standard 11
Mathematics
