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In the curve x= a (cos t + log tan t/2),...

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

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`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`
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