Understanding the Identity
To prove the given trigonometric identity, we start with the left-hand side (LHS) and manipulate it to transform it into the right-hand side (RHS).
Step 1: Simplifying the Left-Hand Side
- LHS = (1 - sec(theta) + tan(theta)) / (1 + sec(theta) - tan(theta))
* Rewrite sec(theta) and tan(theta) in terms of sine and cosine:
* sec(theta) = 1/cos(theta) and tan(theta) = sin(theta)/cos(theta)
- Substitute these into LHS:
* LHS = (1 - (1/cos(theta)) + (sin(theta)/cos(theta))) / (1 + (1/cos(theta)) - (sin(theta)/cos(theta)))
- Combine terms:
* LHS = ((cos(theta) - 1 + sin(theta)) / cos(theta)) / ((cos(theta) + 1 - sin(theta)) / cos(theta))
- Cancel out cos(theta):
* LHS = (cos(theta) - 1 + sin(theta)) / (cos(theta) + 1 - sin(theta))
Step 2: Simplifying the Right-Hand Side
- RHS = (sec(theta) + tan(theta) - 1) / (sec(theta) + tan(theta) + 1)
- Substitute sec(theta) and tan(theta):
* RHS = ((1/cos(theta) + sin(theta)/cos(theta) - 1)) / ((1/cos(theta) + sin(theta)/cos(theta) + 1))
- Combine terms:
* RHS = ((sin(theta) + (1 - cos(theta))) / cos(theta)) / ((sin(theta) + (1 + cos(theta))) / cos(theta))
- Cancel out cos(theta):
* RHS = (sin(theta) + (1 - cos(theta))) / (sin(theta) + (1 + cos(theta)))
Step 3: Establishing Equality
- Both LHS and RHS now appear as:
* LHS = (sin(theta) + (1 - cos(theta))) / (sin(theta) + (1 + cos(theta)))
* RHS = (sin(theta) + (1 - cos(theta))) / (sin(theta) + (1 + cos(theta)))
- Hence, LHS = RHS, confirming the identity.
Conclusion
The given trigonometric identity has been proven by simplifying both sides and showing their equivalence.