If `x=cos t + (log tant)/2, y=sin t,` then find the value of `(d^2y)/(dt^2) and (d^2y)/(dx^2) a tt =pi/4.`

Explanation

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


`frac{d x}{d t}=-sin t+frac{1}{tan (frac{t}{2})} times sec ^{2}(frac{t}{2}) times frac{1}{2}`


`frac{d x}{d t}=-sin t+frac{1}{2 sin (frac{t}{2}) cos (frac{t}{2})}=-sin t+frac{1}{sin t}=frac{cos ^{2} t}{sin t}`


`y=sin t`


`frac{d y}{d t}=cos t`


`frac{d y}{d x}=frac{d y}{d t} times frac{d t}{d x}=cos t times frac{sin ^{2}}{cos ^{2} t}=tan t`


`frac{d^{2} y}{d x^{2}}=sec ^{2} t times frac{d t}{d x}=sec t times frac{sin ^{2} t}{cos ^{2} t}`


`frac{d^{2} y}{d x^{2}}=(sqrt{2})^{2} times frac{sqrt{2}}{frac{1}{2}}=2 sqrt{2}`