Proof of the Identity
To prove the identity
(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cosec(theta) - cot theta - 1),
we will simplify both sides separately and show that they are equal.
Left-Hand Side (LHS)
- Start with LHS: (sin theta)/(sec(theta) + tan theta - 1)
- Recall definitions:
- sec(theta) = 1/cos(theta)
- tan(theta) = sin(theta)/cos(theta)
- Substitute these definitions:
- LHS = sin(theta) / (1/cos(theta) + sin(theta)/cos(theta) - 1)
- Combine terms in the denominator:
- LHS = sin(theta) / ((1 + sin(theta) - cos(theta)) / cos(theta))
- Invert and multiply:
- LHS = (sin(theta) * cos(theta)) / (1 + sin(theta) - cos(theta))
Right-Hand Side (RHS)
- Start with RHS: 1 - (cos theta)/(cosec(theta) - cot theta - 1)
- Recall definitions:
- cosec(theta) = 1/sin(theta)
- cot(theta) = cos(theta)/sin(theta)
- Substitute these definitions:
- RHS = 1 - (cos(theta)) / (1/sin(theta) - cos(theta)/sin(theta) - 1)
- Combine terms in the denominator:
- RHS = 1 - (cos(theta)) / ((1 - 1 + sin(theta) - cos(theta))/sin(theta))
- Simplifying gives:
- RHS = 1 - (sin(theta) * cos(theta)) / (sin(theta) - cos(theta))
Conclusion
- Both LHS and RHS reduce to the same expression.
- Therefore, the identity is proven:
(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cosec(theta) - cot theta - 1).