If `x=10(t-sint)` , `y=12(1-cost)` , find `(dy)/(dx)` .

Explanation

It is given that,
`y=12(1-cos t), x=10(t-sin t)`

Therefore
`frac{d x}{d t}=frac{d}{d t}[10(t-sin t)]=10 cdot frac{d}{d t}(t-sin t)=10(1-cos t) `

`frac{d y}{d t}=frac{d}{d t}[12(1-cos t)]=12 cdot frac{d}{d t}(1-cos t)=12 cdot[0-(-sin t)]=12 sin t `

Therefore
`frac{d y}{d x}=frac{(frac{d y}{d t})}{(frac{d x}{d t})}=frac{12 sin t}{10(1-cos t)}`
`=frac{12 cdot 2 sin frac{t}{2} cdot cos frac{t}{2}}{10 cdot 2 sin ^{2} frac{t}{2}}`

`=frac{6}{5} cot frac{t}{2}`