Understanding the Equation
To solve the equation:
\[ \frac{\csc x}{1+\sec x} + \frac{1+\sec x}{\csc x} = 2 \csc^3 x (\sec x - 1) \]
we will first simplify the left-hand side.

Step 1: Simplifying the Left-Hand Side
- Rewrite the terms using fundamental identities:
- \(\csc x = \frac{1}{\sin x}\)
- \(\sec x = \frac{1}{\cos x}\)
- Substitute these into the equation:
\[ \frac{1/\sin x}{1 + 1/\cos x} + \frac{1 + 1/\cos x}{1/\sin x} \]
- This simplifies to:
\[ \frac{1}{\sin x(1 + 1/\cos x)} + \frac{(1 + 1/\cos x) \sin x}{1} \]
- Combine terms and simplify further.

Step 2: Finding a Common Denominator
- The left-hand side can be expressed with a common denominator of \(\sin x \cos x\):
\[ \frac{\cos x + (1 + \cos x) \sin^2 x}{\sin x \cos x} \]
- This leads to:
\[ \frac{\cos x + \sin^2 x + \sin^2 x \cos x}{\sin x \cos x} \]

Step 3: Simplifying the Right-Hand Side
- The right-hand side can be expressed as:
\[ 2 \csc^3 x (\sec x - 1) = 2 \frac{1}{\sin^3 x} \left(\frac{1}{\cos x} - 1\right) \]
- Simplifying this gives:
\[ 2 \left(\frac{1 - \cos x}{\sin^3 x \cos x}\right) \]

Step 4: Equating Both Sides
- Now, set both simplified sides equal and simplify further:
\[ \frac{(\cos x + \sin^2 x + \sin^2 x \cos x)}{\sin x \cos x} = \frac{2(1 - \cos x)}{\sin^3 x \cos x} \]
- By clearing denominators and simplifying, this should reveal the relationships between \(\sin x\) and \(\cos x\).

Conclusion
- The simplification process leads to a recognition of trigonometric identities, which emphasizes the interrelationships of these functions.
- It is important to practice similar problems for mastery.