Length of cardiod $r=1+\sin\left(\theta\right)$ I'm pretty sure this is correct since I'm able to justify each of the steps but I'm just wondering if there's a more efficient way that isn't just skipping steps? Because I had to do 2 substitutions, one of which converted sine into cosine.
\begin{align*}
\text{Length }= & \ 2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{2 + 2\sin\left(\theta\right)} d\theta \\
= & \ 2  \sqrt{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{1 + \sin\left(\theta\right)} d\theta\\
= & \ - 2\sqrt{2} \int_{\pi}^{0} \sqrt{1+ \sin\left(\frac{\pi}{2}-x\right) } dx \\
= & \ - 2\sqrt{2} \int_{\pi}^{0} \sqrt{1+ \cos\left(x\right) } dx \\
= & \ - 2\sqrt{2} \int_{\pi}^{0} \sqrt{2 \cos^2\left(\frac{x}{2}\right)} dx \\
= & \ - 4 \int_{\pi}^{0} \left|\cos\left(\frac{x}{2}\right)\right| dx \\
= & \ 4 \int_{0}^{\pi} \cos\left(\frac{x}{2}\right) dx \\
= & \ 4 \cdot 2 \int_{0}^{\frac{\pi}{2}} \cos\left(u\right) du \\
= & \ 8 \left[\sin\left(u\right)\right]_{u=0}^{u=\frac{\pi}{2}} \\
= & \ 8 \left[\sin\left(\frac{\pi}{2}\right) - \sin\left(0\right) \right] \\
= & \ 8 \\
\end{align*}
edit: admittedly, I do know of other answers but I have no idea how one would have come across such trigonometric identities, or reason out how such an identity would be beneficial.

Everything is correct including the setup of the integral (using symmetry), but note that $1 + \sin x = (\cos x/2 + \sin x/2)^2$ as mentioned here.
When taking the square root you have $|\cos x/2 + \sin x/2|$, but since this equals $|\sqrt{2} \cos(x/2 - \pi/4)|$ which is non-negative for $-\frac{\pi}{2} ≤ x ≤ \frac{3 \pi}{2}$, $|\cos x/2 + \sin x/2| = \cos x/2 + \sin x/2$ for your integral.
