Given,
x = a (cos t + t sin t)
Differentiating both sides with respect to t, we get
\(\frac{dx}{dt}\) = a (-sin t + t cos t + sin t)
⇒ \(\frac{dx}{dt}\) = at cos t ...(i)
Differentiating again with respect to t, we get
\(\frac{d^2x}{dt^2}\) = a (- t sin t + t cos t)
= a (cos t + t sin t)
Again,
y = a (sin t - cos t)
Differentiating with respect to t, we get
\(\frac{dy}{dt}=\) a (cos t - t sin t - cos t) ...(ii)
⇒ \(\frac{dy}{dt}=\) at sin t
Differentiating again with respect to t, we get
\(\frac{d^2y}{dt^2}\) = a (t cos t +sin t)
Now,
\(\frac{dy}{dx}=\) \(\frac{dt/dy}{dt/dy}\)
[from (i) and (ii)]
⇒ \(\frac{dy}{dx}=\) \(\frac{at\,sin\,t}{at\,cos\,t}\)
⇒ \(\frac{dy}{dx}=\) tan t
Differentiating again with respect to x, we get
\(\frac{d^2y}{dx^2}=\) sec2 t.\(\frac{dt}{dx}\)