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If $x=\cos1^{\circ} \cdot \cos2^{\circ} \cdot \cos3^{\circ}\dots\cos89^{\circ}$ and $y=\cos2^{\circ}\cos6^{\circ}\cos10^{\circ}\dots\cos86^{\circ}$ then what the integer is nearest to $\dfrac{2}{7}\log _{2} \left( \dfrac{y}{x}\right )$is:

  1. $19$
  2. $17$
  3. $15$
  4. $21$
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Answer is A

$x= \frac{1}{2^{44}}[(2cos1.cos89)(2cos2.cos88)(2cos3.cos87)....(2cos44.cos46).cos45]$

Applying 2cosA.cosB=cos(A+B)+cos(A-B)

$x= \frac{1}{2^{44}}[(cos90+cos88)(cos90+cos86)(cos90+cos84)....(cos90+cos2)cos45]$

$x= \frac{1}{2^{44}}[(0+cos88)(0+cos86)(0+cos84)....(0+cos2).\frac{1}{\sqrt{2}}] $

$x=\frac{1}{2^{44}+2^\frac{1}{2}}[cos88.cos86.cos84.......cos2)]$

$x=\frac{1}{2^{44}+2^{22}+2^\frac{1}{2}}[(2cos88cos2).(2cos86cos4).(2cos84cos6).......] $

$x=\frac{1}{2^{66}+2^\frac{1}{2}}[(cos90+cos86).(cos90+cos82).(cos90+cos78).......] $

$x=\frac{1}{2^\frac{133}{2}}[(cos86).(cos82).(cos78)......(cos6).(cos2)] =\frac{1}{2^\frac{133}{2}}*y$

$\frac{y}{x}={2^\frac{133}{2}}$

$log\frac{y}{x}=log {2^\frac{133}{2}}$

$log_{2}\frac{y}{x}=\frac{133}{2}$

$\frac{2}{7}log_{2}\frac{y}{x}=\frac{133}{2}*\frac{2}{7}=19$

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