Understanding the Equation
The equation given is:
Sin - Cos + 1 / Sin + Cos - 1 = 1 / Sec - Tan
This can be simplified by using trigonometric identities.
 

Step 1: Simplifying the Left Side
- Start with the left side:
Sin - Cos + 1 / Sin + Cos - 1
- Rewrite it as:
(Sin - Cos + 1) / (Sin + Cos - 1)
 

Step 2: Analyzing the Right Side
- The right side is:
1 / Sec - Tan
- Recall that:
Sec = 1 / Cos
Tan = Sin / Cos
Thus,
1 / Sec - Tan = Cos / (1 - Sin)
 

Step 3: Equating Both Sides
- Set both sides equal:
(Sin - Cos + 1) / (Sin + Cos - 1) = Cos / (1 - Sin)
 

Step 4: Cross-Multiplying
- Cross-multiply to eliminate the fractions:
(Sin - Cos + 1)(1 - Sin) = Cos(Sin + Cos - 1)
 

Step 5: Expanding and Simplifying
- Expand both sides:
Sin - Sin^2 - Cos + CosSin + 1 - Sin = CosSin + Cos^2 - Cos
- Combine like terms and simplify.
 

Conclusion
Through careful manipulation and simplification of both sides, we confirm that the original equation holds true. This process illustrates the importance of understanding trigonometric identities and algebraic manipulation in solving equations.