Write the correct alternative in the following:
If $x=f(t) \cos t-f^{\prime}(t) \sin t$ and $y=f(t) \sin t+f^{\prime}(t) \cos t, t h e n\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}=$
A. $f(t)-f^{\prime \prime}(t)$
B. $\left\{f(t)-f^{\prime \prime}(t)\right\}^{2}$
C. $\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$
D. none of these
Given:
$x=f(t) \cos t-f^{\prime}(t) \sin t$
$y=f(t) \sin t+f^{\prime}(t) \cos t$
$\frac{d x}{d t}=f^{\prime}(t) \cos t-f(t) \sin t-f^{\prime}(t) \sin t-f^{\prime}(t) \cos t$
$=-f(t) \sin t-f^{\prime \prime}(t) \sin t$
$=-\sin t\left[f(t)+f^{\prime \prime}(t)\right]$
$\left(\frac{d x}{d t}\right)^{2}=\left\{-\sin t\left[f(t)+f^{\prime}(t)\right]\right\}^{2}$
$=(\sin t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$
$\frac{d y}{d t}=f^{\prime}(t) \sin t+f(t) \cos t+f^{\prime \prime}(t) \cos t-f^{\prime}(t) \sin t$
$=f(t) \cos t+f^{\prime \prime}(t) \cos t$
$=\cos t\left[f(t)+f^{\prime \prime}(t)\right]$
$\left(\frac{d y}{d t}\right)^{2}=\left\{\cos t\left[f(t)+f^{\prime \prime}(t)\right]\right\}^{2}$
$=(\cos t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$
$\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}=(\sin t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}+(\cos t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$
$=\left\{\mathrm{f}(\mathrm{t})+\mathrm{f}^{\prime \prime}(\mathrm{t})\right\}^{2}$
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