When doing some trig practice in my free time, I came across the problem if $\displaystyle \sin 2\alpha = \frac{1}{7}$, find $\displaystyle \sin^4 \alpha + \cos^4 \alpha$.

I did a little research online, and I found that $\displaystyle \sin^4 \alpha + \cos^4 \alpha = \frac{1}{4}(\cos 4\alpha + 3)$, which simplified the problem greatly, and I was able to find that the answer is $\displaystyle \frac{97}{98}$. Nevertheless, I feel that there is probably a "cleaner" way of doing this that does not require knowing the above equivalence.

Does anyone have any suggestions on how to attempt this question? Thanks!