Need Help for proofs in the Basel problem.

Question

My teacher told me to prove that $$\tan ^{−1} \frac {(1−)}{\sqrt{(1−^2)}}$$ leads to $$\tan^2(\theta) =\frac {(1 − u)}{(1+ u)}$$ and $$\sec^2(\theta) = \frac {2}{(1+u)}$$ using trigonometric identities.

I was able the prove the $\tan^2 \theta$ one but I am stuck with the $\sec^2 \theta$ one.

I also need to prove that $$u = 2\cos^2(\theta) − 1 = \cos(2 \theta)$$

Context in the image.Basel Problem Thank You !

Answer 1

Hint:

Use the trigonometric identity $$\sec^2 (\theta) - \tan^2 (\theta) = 1$$ and rearrange; use the results you got from $\tan^2 (\theta)$ to get $\sec^2 (\theta).$ Then, use the result for $\sec^2 (\theta)$ and rearrange to find $u$.