`"If "x=sqrt(a^(sin-1_(t))),y=sqrt(a^(cos^(-1_(t)))),agt0 and -1 lt t lt 1," show that "(dy)/(dx)=-(y)/(x).`

Explanation

`x = sqrt(a^("sin"^(-1)t))`
`(dx)/(dt) = (1)/(2sqrt(a^("sin"^(-1)t))) * a^("sin"^(-1)t) * "log"a * (1)/(1-t^(2))`
`= (1)/(2)sqrt(a^("sin"^(-1)t)) * "log"a * (1)/(sqrt(1-t^(2)))`
`= (x)/(2) * "log"a * (1)/(sqrt(1-t^(2)))`
`"Now " y= sqrt(a^("cos"^(-1)t))`
`rArr (dy)/(dt) = (1)/(2sqrt(a^("cos"^(-1)t))) * a^("cos"^(-1)t) * "log"a * ((1))/sqrt(1-t^(2))`
`= (-1)/(2) * sqrt(a^("cos"^(-1)t)) * "log"a * (1)/sqrt(1-t^(2))`
`= (-y)/(2) * "log"a * (1)/sqrt(1-t^(2))`
`therefore (dy)/(dx) = (dy//dt)/(dx//dt) = ((-y)/(2) "log" a * (1)/(sqrt(1-t^(2))))/((x)/(2)"log"a * (1)/(sqrt(1-t^(2)))) = (-y)/(x)`