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The subtangent is a segment ST on the X-axis from point S, an abscissa of some point P on a curve, to point T, intersection of a tangent to a curve at point P with the X-axis.
Knowing parametric expressions for x=f(t) and y=g(t), for any parameter t we know abscissa and ordinate of a point P(t) on a curve.
In this case we will use ordinate y=a(1-cos t) and will determine tan(Psi). That will be sufficient to find ST.
To calculate tan(Psi)=dy/dx we will use the property of parametric curve:
dy/dx = (dy/dt)/(dx/dt)
Using this approach,
tan(Psi) = dy/dx = [a(sin t)]/[a(1+cos t)] = sin t/(1+cos t)
Now we can calculate ST:
ST = PS*tan(Psi) = y*dy/dx = a(1-cos t)*sin t/(1+cos t)
This does not resemble any of the answers provided.
However, if x=a(t-sin t) then dx/dt = 1-cos t and answer asin t would be correct.
