Question

Assertion :If $exp\left\{\right\{si{n}^{2}x+si{n}^{4}x+si{n}^{6}x+.....inf\left\}lo{g}_{e}2\right\}$ satisfies the equation $x^2-9x+8=0$, then the value of $\frac{cosx}{cosx+sinx}$ is $\frac{\sqrt{3}-1}{2}(0<x<\pi /2)$ Reason: $si{n}^{2}x+si{n}^{4}x+si{n}^{6}x+....$ $\infty =se{c}^{2}x$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$Exp\left\{\left({\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+....\infty \right){\log }_{e}2\right\}=e^{\frac{{\sin }^{2}x}{1-{\sin }^{2}x}{\log }_{e}2}=e^{\frac{{\sin }^{2}x}{{\cos }^{2}x}{\log }_{e}2}$
$\Rightarrow 2^{{\tan }^{2}x}$ satisfy $x^2-9x+8=0$
$\Rightarrow x=1,8$
$\therefore 2^{{\tan }^{2}x}=1$ and $2^{{\tan }^{2}x}=8$
$\Rightarrow {\tan }^{2}x=0$ and ${\tan }^{2}x=3$
$\Rightarrow x=n\pi$ and ${\tan }^{2}x={\left(\tan \frac{\pi }{3}\right)}^2$
$\Rightarrow x=n\pi$ and $x=n\pi \pm \frac{\pi }{3}$
Neglecting $x=n\pi$ as $\Rightarrow x=\frac{\pi }{3}\in (0,\frac{\pi }{2})$
$\therefore \frac{\cos x}{\cos x+\sin x}=\frac{\frac{1}{2}}{\frac{1}{2}+\frac{\sqrt{3}}{2}}=\frac{1}{1+\sqrt{3}}=\frac{\sqrt{3}-1}{3-1}=\frac{\sqrt{3}-1}{2}$