If x = cos t and y = sin t, what is d^2y/dx^2?

Question 81

HardOriginal question

If $x = \cos t$ and $y = \sin t$ , what is $\frac{d^2y}{dx^2}$ ?

\operatorname{cosec}^{2} t

Correct answerYour answer

- \cot t

Correct answerYour answer

\operatorname{cosec}^{3} t

Correct answerYour answer

- \operatorname{cosec}^{3} t

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\frac{dy}{dx} = \frac{\cos t}{-\sin t} = -\cot t$ .
$\frac{d}{dt}(-\cot t) = \operatorname{cosec}^2 t$ , and dividing by $\frac{dx}{dt} = -\sin t$ gives $\frac{d^2y}{dx^2} = -\operatorname{cosec}^3 t$ .

Worth noting

$\operatorname{cosec}^2 t$ is the derivative with respect to $t$ , not yet divided by $\frac{dx}{dt}$ .

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