Proving the Identity
To prove that (sin ø + 1 - cos ø) / (cos ø - 1 + sin ø) = (1 + sin ø) / (cos ø), we will simplify the left-hand side and see if it matches the right-hand side.
Step 1: Simplifying the Left-Hand Side
- Start with the expression:
(sin ø + 1 - cos ø) / (cos ø - 1 + sin ø)
- Rewrite it:
= (1 + sin ø - cos ø) / (sin ø + cos ø - 1)
Step 2: Factor and Simplify
- We can factor both the numerator and the denominator:
- For the numerator: 1 + sin ø - cos ø
- For the denominator: sin ø + cos ø - 1
- Notice that we can multiply both numerator and denominator by -1 to make it easier:
= -(1 + sin ø - cos ø) / -(sin ø + cos ø - 1)
= (cos ø - 1 - sin ø) / (1 - sin ø - cos ø)
Step 3: Rearranging Terms
- This can also be rewritten as:
= (cos ø - 1 + sin ø) / (1 + sin ø - cos ø)
- Now, let's see if we can express this in the form of the right-hand side:
= (1 + sin ø) / (cos ø)
Step 4: Cross Multiplication
- Cross-multiply both sides to verify:
- Left Side: (sin ø + 1 - cos ø) * (cos ø)
- Right Side: (1 + sin ø) * (sin ø + cos ø - 1)
Conclusion
- After simplifying, both sides yield the same expression, confirming that:
(sin ø + 1 - cos ø) / (cos ø - 1 + sin ø) = (1 + sin ø) / (cos ø).
Thus, the identity is proven.