Arc Length Of Parametric Curve

Question

I attached the problem as a file: $$x=e^{-t} \cos t, \, y= e^{-t} \sin t, \, 0 \le t \le \frac{\pi}{2}$$ $$\frac{dx}{dt}= -e^{-t}(\sin t + \cos t), \, \frac{dy}{dt} = e^{-t}(\cos t - \sin t)$$ \begin{align*} s & = \int_0^{\pi / 2} \sqrt{\left ( \frac{dx}{dt} \right ) ^2 + \left ( \frac{dy}{dt} \right )^2} \, dt\\ & = \int_0^{\pi / 2} \sqrt{2e^{-2t}} \, dt = - \sqrt{2} \int_0^{\pi / 2} e^{-t}(-1) \, dt\\ & \phantom{= \int_0^{\pi / 2} \sqrt{2e^{-2t}} \, dt \,} = \left [ - \sqrt{2}e^{-t} \right ]_0^{\pi / 2}\\ & \phantom{= \int_0^{\pi / 2} \sqrt{2e^{-2t}} \, dt \,} = \sqrt{2} (1 - e^{-\pi / 2}) \approx 1.12 \end{align*}

Where did the trig functions go? I sifted through the different trig identities and formulas, but couldn't find anything that I could use. What should I do?

Answer 1

$$ \begin{align} & (\sin(t)+\cos(t))^2+(\sin(t)-\cos(t))^2 \\ & = \sin^2(t)+\cos^2(t)+2\cos(t)\sin(t)+\sin^2(t)+\cos^2(t)-2\cos(t)\sin(t) \\ & =2(\sin^2(t)+\cos^2(t)) \\ & =2 \end{align}$$