Find the lengths of sub tangent, subnormal at a point $t$ on the curve $x=a(\cos t+t\sin t),y=a(\sin t-t\cos t)$

Given curves are  $x=a(\cos t+t\sin t); y=a(\sin t-t\cos t)$

diff w.r. to $\text{'}t\text{'}$

$\frac{dx}{dt}=a[-\sin t+t\cos t+\sin t]=\text{ at }\cos t$

$\frac{dy}{dt}=a[\cos t-(t(-\sin t)+\cos t\left)\right]$

$=a[\cos t+t\sin t-\cos t]=\text{ at sint }$

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{at\sin t}{at\cos t}=\tan t$

Length of sub tangent $=\left|\frac{y}{m}\right|$

$=\left|\frac{a(\sin t-t\cos t)}{\tan t}\right|$

Length of sub normal $=\left|ym\right|$

$=\left|a\right(\sin t-t\cos t)\tan t|$