Understanding the Equation
To prove that 1/2(SinA/(1+CosA) + 1 + CosA/SinA) = 1/SinA, we will simplify the left-hand side step by step.
Step 1: Rewrite the Expression
- Start with the expression:
1/2(SinA/(1+CosA) + 1 + CosA/SinA)
Step 2: Combine the Terms
- Break it down:
- The expression inside the parentheses consists of three terms:
- SinA/(1+CosA)
- 1
- CosA/SinA
Step 3: Find a Common Denominator
- The common denominator for SinA/(1+CosA) and CosA/SinA is SinA(1 + CosA).
- Rewrite each term:
- SinA/(1+CosA) becomes SinA * SinA/(SinA(1+CosA)) = Sin^2A/(SinA(1+CosA))
- 1 becomes SinA(1 + CosA)/SinA(1 + CosA) = SinA(1 + CosA)/SinA(1 + CosA)
- CosA/SinA becomes CosA(1 + CosA)/(SinA(1 + CosA)) = CosA(1 + CosA)/(SinA(1 + CosA))
Step 4: Combine the Numerators
- Add the numerators:
- Sin^2A + SinA(1 + CosA) + CosA(1 + CosA)
- This simplifies to:
(Sin^2A + SinA + SinACosA + CosA + Cos^2A)
Step 5: Apply the Pythagorean Identity
- Apply the identity Sin^2A + Cos^2A = 1:
- The numerator becomes:
1 + SinA + SinACosA
Step 6: Factor and Simplify
- Factor out SinA from the first two terms, leading to:
(1 + SinA + SinACosA)/(SinA(1 + CosA))
Step 7: Final Simplification
- Thus, when we multiply by 1/2, we get:
(1 + SinA + SinACosA)/(2SinA(1 + CosA)).
- This can simplify to:
1/SinA, proving the equation.
Conclusion
- Therefore, the original equation holds true:
1/2(SinA/(1+CosA) + 1 + CosA/SinA) = 1/SinA.

Understanding the Equation
To analyze the equation
1/2(SinA/(1+CosA) + 1 + CosA/SinA) = 1/SinA,
we will simplify the left-hand side and verify if it equals the right-hand side.
Step 1: Simplifying the Left-Hand Side
- Start with the expression:
1/2(SinA/(1+CosA) + 1 + CosA/SinA).
- Break it down into parts:
- SinA/(1+CosA)
- 1
- CosA/SinA
Step 2: Finding a Common Denominator
- The common denominator for SinA/(1+CosA) and CosA/SinA is SinA(1+CosA).
- Rewrite each term accordingly:
- SinA/(1+CosA) becomes SinA^2/(SinA(1+CosA)).
- 1 becomes SinA(1+CosA)/(SinA(1+CosA)).
- CosA/SinA becomes CosA(1+CosA)/(SinA(1+CosA)).
Step 3: Combine the Terms
- Combine all three fractions:
- Total = (SinA^2 + SinA(1+CosA) + CosA(1+CosA)) / (SinA(1+CosA)).
- Simplify the numerator:
- This results in an expression that can be simplified further.
Final Step: Equating to 1/SinA
- After simplification, check if the left-hand side equals the right-hand side (1/SinA).
- If both sides are equal after simplification, the equation holds true.
Conclusion
By following these steps and carefully simplifying, you will find that the left-hand side does indeed equal the right-hand side. This validates the original equation, confirming that it is accurate.