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Question
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
Solution
x = sin–1(et), y = `sqrt(1 - e^(2t))`
Differentiating x and y w.r.t. t, we get
`dx/dt = d/dt[sin^-1(e^t)]`
= `(1)/sqrt(1 - (e^t)^2).d/dt(e^t)`
= `(1)/sqrt(1 - e^(2t)) xx e^t = e^t/sqrt(1 - e^(2t))` and
`dy/dt = d/dt(sqrt(1 - e^(2t)))`
= `(1)/(2sqrt(1 - e^(2t))).d/dt(1 - e^(2t))`
= `(1)/(2sqrt(1 - e^(2t))). xx (0 - e^(2t) xx 2)`
= `(-e^(2t))/sqrt(1 - e^(2t))`
∴ `dy/dx = ((dy/dt))/((dx/dt)`
= `(((-e^(2t))/sqrt(1 - e^(2t))))/(((e^t)/sqrt(1 - e^(2t)))`
= – et
= – sin x ...[∵ x = sin–1(et)]
∴ `sin x + dy/dx` = 0.
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