In the curve `x= a (cos t+ log tan(t/2))`,` y =a sin t`. Show that the portion of the tangent between the point of contact and the x-axis is of constant length.

Explanation

`x=a("cos"t + "log tan"(t)/(2)) " and "y=a"sin"t`
`rArr (dx)/(dt)=a[-"sin"t + ("sec"^(2)(t)/(2))/(2"tan"(t)/(2))] "and" (dy)/(dx) = a"cos"t`
`=a[-"sin"t + (1)/(2"tan"(t)/(2) "cos" (t)/(2))]`
`=a(-"sin"t + (1)/("sin"t))`
`=a((1-"sin"^(2)t)/("sin"t))= a("cos"^(2)t)/("sin"t)`
`"Now", (dy)/(dx) = (dy//dt)/(dx//dt) = ((a"cos"t)/(a"cos"^(2)t))/("sin"t) = "tan"t`