Problem (Algebra):
Let T be a rational number. Compute $\sin ^{2} \frac{T \pi}{2}+\sin ^{2} \frac{(5-T) \pi}{2}$.

Solution:
Note that $\sin \frac{(5-T) \pi}{2}=\cos \left(\frac{\pi}{2}-\frac{(5-T) \pi}{2}\right)=\cos \left(\frac{T \pi}{2}-2 \pi\right)=\cos \frac{T \pi}{2}$. Thus the desired quantity is $\sin ^{2} \frac{T \pi}{2}+\cos ^{2} \frac{T \pi}{2}=\mathbf{1}$ (independent of $T$ ).

Final answer: 1