I’ve gotten up to this point, I feel like there is some ending to it, but I can’t figure it out. Any help will be appreciated!
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As you want the last has limit \frac{1}{2}, then the limit of \left(\tan\left(\frac{\alpha}{2}\right)\right)^{2m} is 0 so \left|\tan\left(\frac{\alpha}{2}\right)\right|<1. From here you can find the values of \alpha.