If [e^( ^2x + ^4x + ^6x + ... ) _e2] satisfies the equation t2 - 9t + 8 = 0, then the valu…

Question

If
$${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$$
satisfies the equation t2 - 9t + 8 = 0, then the value of

$${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi  \over 2}} \right)$$ is :

Accepted Answer

$${e^{({{\cos }^2}x + {{\cos }^4}x + ...........\infty )\ln 2}} = {2^{{{\cos }^2}x + {{\cos }^4}x + ...........\infty }}$$

= $${2^{{{{{\cos }^2}x} \over {1 - {{\cos }^2}x}}}}$$

$$ = {2^{{{\cot }^2}x}}$$

Given, $${t^2} - 9t + 8 = 0 \Rightarrow t = 1,8$$

$$ \Rightarrow {2^{{{\cot }^2}x}} = 1,8 \Rightarrow co{t^2}x = 0,3$$

$$0 < x < {\pi  \over 2} \Rightarrow \cot x = \sqrt 3 $$

$$ \therefore $$ $$  {{2\sin x} \over {\sin x + \sqrt 3 \cos x}} = {2 \over {1 + \sqrt 3 \cot x}} = {2 \over 4} = {1 \over 2}$$

Solutions · 1