Equations $y = 2A \cos ^2 \omega \,t$ and $y = A (\sin \omega t + \cos \omega t )$ represent the motion of two particles.
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$y=2 A \cos ^{2} \omega t$
$\Rightarrow y=A(\cos 2 \omega t+1)$
$\Rightarrow y-A=A \cos 2 \omega t$
$\Rightarrow A_{1}=A ; \omega_{1}=2 \omega$
$y=A(\sin \omega t+\sqrt{3} \cos \omega t)$
$\Rightarrow y=2 A\left(\frac{1}{2} \sin \omega t+\frac{\sqrt{3}}{2} \cos \omega t\right)$
$\Rightarrow y=2 A \sin \left(\omega t+\frac{\pi}{3}\right)$
$\Rightarrow A_{2}=2 A ; \omega_{2}=\omega$
Ratio of maximum speeds $=\frac{A_{1} \omega_{1}}{A_{2} \omega_{2}}=\frac{A(2 \omega)}{(2 A) \omega}=\frac{1}{1}$
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