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If $\mathrm{y}=\cos ^{2}\left(\frac{5 x}{2}\right)-\sin ^{2}\left(\frac{5 x}{2}\right)$, then $\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} x^{2}}\right)=$
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Verified Answer
The correct answer is:
$-25 y$
Given
$$
\begin{aligned}
y &=\cos ^{2}\left(\frac{5 x}{2}\right)-\sin ^{2}\left(\frac{5 x}{2}\right) \\
y &=\cos \left(2 \times \frac{5 x}{2}\right) \Rightarrow y=\cos 5 x \\
\therefore \frac{d y}{d x} &=-5 \sin 5 x \Rightarrow \frac{d^{2} y}{d x^{2}} \\
&=-25 \cos 5 x=-25 y
\end{aligned}
$$
$$
\begin{aligned}
y &=\cos ^{2}\left(\frac{5 x}{2}\right)-\sin ^{2}\left(\frac{5 x}{2}\right) \\
y &=\cos \left(2 \times \frac{5 x}{2}\right) \Rightarrow y=\cos 5 x \\
\therefore \frac{d y}{d x} &=-5 \sin 5 x \Rightarrow \frac{d^{2} y}{d x^{2}} \\
&=-25 \cos 5 x=-25 y
\end{aligned}
$$
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