Question

X= a ( cost +logtan t/2 y = asin t find dy / dx​

Answer 1

Answer:
`(dy)/(dx) = \tan t`

Step-by-step explanation:

`x = a(\cos t + \log \tan\frac{t}2)`

Differentiate with respect to t,

`(dx)/(dt) =a \frac{d{(\cos t + \log\tan \frac{t}2)}}{dt}`

`= a[(d(\cos t))/(dt) + \frac{d(\log\tan \frac{t}2)}{dt}]`

`= a[ -\sin t + \frac{1}{\tan  \frac{t}2 }* sec^2\frac{t}2* \frac12]`

`= a [ -\sin t + \frac12 \frac{1}{\sin \frac{t}2\cdot\cos \frac{t}2 }]`

Since
`2\sin A\cdot\cos A = \sin2 A`

`= a [ -\sin t + \frac1{\sin t}]`

`= a ( 1 - \sin^2 t)/(\sin t)`

`= a(\cos^2t)/(\sin t)`

`= a\cot t\cdot \cos t .......(1)`

again,
`y = a\sin t`

`(dy)/(dt) = a\cos t .......(2)`

now,
`(dy)/(dx) =\frac {(dy)/(dt)}{(dx)/(dt)}` [from (1) and (2)]

`=\frac {a\cos t}{a\cot t.\cos t}`

`= (1)/(\cot t )`

Hence,
`(dy)/(dx) = \tan t`