Để rút gọn các biểu thức, ta sẽ sử dụng các công thức lượng giác cơ bản.

a) \( A = \sin^2 \alpha + \sin^2 \alpha \tan^2 \alpha \)

Ta có: \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\)

Do đó:
\[
\tan^2 \alpha = \left(\frac{\sin \alpha}{\cos \alpha}\right)^2 = \frac{\sin^2 \alpha}{\cos^2 \alpha}
\]

Thay vào biểu thức \(A\):
\[
A = \sin^2 \alpha + \sin^2 \alpha \cdot \frac{\sin^2 \alpha}{\cos^2 \alpha} = \sin^2 \alpha + \frac{\sin^4 \alpha}{\cos^2 \alpha}
\]

Ta có: \(\sin^2 \alpha = 1 - \cos^2 \alpha\)

Do đó:
\[
A = \sin^2 \alpha + \frac{\sin^4 \alpha}{\cos^2 \alpha} = \sin^2 \alpha + \frac{(1 - \cos^2 \alpha)^2}{\cos^2 \alpha}
\]

b) \( B = \frac{2 \sin^2 \alpha - 1}{\sin^2 \alpha - \sin \alpha \cos \alpha} \)

Ta có: \( \sin^2 \alpha = 1 - \cos^2 \alpha \)

Do đó:
\[
B = \frac{2 (1 - \cos^2 \alpha) - 1}{(1 - \cos^2 \alpha) - \sin \alpha \cos \alpha} = \frac{2 - 2 \cos^2 \alpha - 1}{1 - \cos^2 \alpha - \sin \alpha \cos \alpha} = \frac{1 - 2 \cos^2 \alpha}{1 - \cos^2 \alpha - \sin \alpha \cos \alpha}
\]

c) \( C = \sin^2 \alpha \cos^2 \alpha + \cos^2 \alpha + \sin^4 \alpha \)

Ta có: \(\sin^2 \alpha + \cos^2 \alpha = 1\)

Do đó:
\[
C = \sin^2 \alpha \cos^2 \alpha + \cos^2 \alpha + \sin^4 \alpha = \cos^2 \alpha (\sin^2 \alpha + 1) + \sin^4 \alpha = \cos^2 \alpha (1) + \sin^4 \alpha = \cos^2 \alpha + \sin^4 \alpha
\]

d) \( D = \frac{1 - \cos \alpha}{\sin^2 \alpha} - \frac{1}{1 + \cos \alpha} \)

Ta có: \(\sin^2 \alpha = 1 - \cos^2 \alpha\)

Do đó:
\[
D = \frac{1 - \cos \alpha}{1 - \cos^2 \alpha} - \frac{1}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{(1 - \cos \alpha)(1 + \cos \alpha)} - \frac{1}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{1 - \cos^2 \alpha} - \frac{1}{1 + \cos \alpha}
\]

Rút gọn tiếp:
\[
D = \frac{1 - \cos \alpha}{(1 - \cos \alpha)(1 + \cos \alpha)} - \frac{1}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{1 - \cos^2 \alpha} - \frac{1}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{\sin^2 \alpha} - \frac{1}{1 + \cos \alpha}
\]

Vậy các biểu thức đã được rút gọn như sau:
a) \( A = \sin^2 \alpha + \frac{\sin^4 \alpha}{\cos^2 \alpha} \)
b) \( B = \frac{1 - 2 \cos^2 \alpha}{1 - \cos^2 \alpha - \sin \alpha \cos \alpha} \)
c) \( C = \cos^2 \alpha + \sin^4 \alpha \)
d) \( D = \frac{1 - \cos \alpha}{\sin^2 \alpha} - \frac{1}{1 + \cos \alpha} \)