The tangent to the cycloid x=a(t- sin t), y=a(1- cos t)

The tangent to the cycloid $x=a(t-\sin t),\ y=a(1-\cos t)$ drawn at an arbitrary point $M$ intersects the coordinate axes at points $S$ (on $Oy$) and $T$ (on $Ox$). Form the ratio $\dfrac{\overline{SM}^{\,2}}{\overline{ST}^{\,2}}$ as a function of $t$, and find its limit as $t\to0$.

Step 1: Slope of the tangent at $M$

With $\dot x=a(1-\cos t)$, $\dot y=a\sin t$, and the half-angle identities $\sin t=2\sin\tfrac t2\cos\tfrac t2$, $1-\cos t=2\sin^2\tfrac t2$: \[k = \frac{dy}{dx} = \frac{\sin t}{1-\cos t} = \cot\frac t2. \tag{1}\]

Step 2: The $y$-intercept $S$

Setting $x=0$ in the tangent $y-a(1-\cos t)=k[x-a(t-\sin t)]$, and simplifying with the half-angle forms (the $\sin^2$ and $\cos^2$ terms combine to $2a$): \[y_{S} = a(1-\cos t) - ka(t-\sin t) = 2a - \frac{at\cos\tfrac t2}{\sin\tfrac t2}. \tag{2}\]

Step 3: The $x$-intercept $T$

Setting $y=0$ and simplifying similarly (using $\sin t=2\sin\tfrac t2\cos\tfrac t2$ to cancel a common factor): \[x_{T} = a(t-\sin t) - a(1-\cos t)\tan\frac t2 = a\!\left(t-2\tan\frac t2\right). \tag{3}\]

Step 4: The two chord lengths

Since $M$, $S$, $T$ all lie on the same line of slope $k=\cot\tfrac t2$, each squared distance is $(\Delta x)^{2}(1+k^{2})$, and $1+\cot^2\tfrac t2 = 1/\sin^2\tfrac t2$: \[\overline{SM}^{\,2} = a^{2}(t-\sin t)^{2}\!\left(1+\cot^{2}\frac t2\right) = \frac{a^{2}(t-\sin t)^{2}}{\sin^{2}\tfrac t2}, \tag{4}\]

\[ \overline{ST}^{\,2} = a^{2}\!\left(t-2\tan\frac t2\right)^{2}\!\left(1+\cot^{2}\frac t2\right) = \frac{a^{2}\left(t-2\tan\tfrac t2\right)^{2}}{\sin^{2}\tfrac t2}. \tag{5} \]

Step 5: The ratio

\[ \frac{\overline{SM}^{\,2}}{\overline{ST}^{\,2}} = \left(\frac{t-\sin t}{t-2\tan\tfrac t2}\right)^{2}. \tag{6} \]

Step 6: Limit as $t\to0$

By Taylor expansion, $t-\sin t \approx \dfrac{t^{3}}{6}$ and $2\tan\tfrac t2 \approx t+\dfrac{t^{3}}{12}$, so $t-2\tan\tfrac t2\approx-\dfrac{t^{3}}{12}$; hence

\[ \frac{t-\sin t}{t-2\tan\tfrac t2} \;\longrightarrow\; \frac{1/6}{-1/12} = -2, \qquad \lim_{t\to0}\frac{\overline{SM}^{\,2}}{\overline{ST}^{\,2}} = (-2)^{2} = 4. \tag{7} \]

Step 7: Verification

\[t=0.5\text{: ratio (6)} \approx3.70853,\ \text{direct coordinate computation} \approx3.70853\ \checkmark\]

\[t=0.01\text{: ratio} \approx3.99988\ \text{(approaching 4)}\ \checkmark;\qquad t=0.001\text{: ratio} \approx3.999999\ \checkmark\]