Question

The parametric equations of a curve are
x = e^−t cos t y = e^−t sin t.

Show that dy/dx= tan (t −1/4π)

Answer 1

@mathmale

Answer 2

I differentiate both side with respect to t and got \[dy/dx = \frac{ e ^{-t }\cos t-e ^{-t}\sin t }{ -e ^{-t}\sin t-e ^{-t}\cos t }\]

Answer 3

@ganeshie8 please help

Answer 4

\[\frac{dy}{dx} = \frac{ e ^{-t }\cos t-e ^{-t}\sin t }{ -e ^{-t}\sin t-e ^{-t}\cos t }=\frac{\sin t-\cos t }{ \sin t+\cos t }\]
\[\begin{align*}
\frac{\sin t-\cos t}{\sin t+\cos t}&=\frac{\dfrac{\sin t}{\cos t}-\dfrac{\cos t}{\cos t}}{\dfrac{\sin t}{\cos t}+\dfrac{\cos t}{\cos t}}\\\\
&=\frac{\tan t-1}{1+\tan t}\\\\
&=\frac{\tan t-\tan\dfrac{\pi}{4}}{1+\tan t\tan\dfrac{\pi}{4}}\\\\
&=\tan\left(t-\frac{\pi}{4}\right)
\end{align*}\]