Proving the Identity
To prove the identity: (sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cos cos theta - cot theta - 1), we will simplify both sides step by step.
Step 1: Simplifying the Left Side
- Start with the left side: (sin theta)/(sec(theta) + tan theta - 1).
- Recall that sec(theta) = 1/cos(theta) and tan(theta) = sin(theta)/cos(theta).
- Substitute these definitions:
- Left Side = (sin theta) / (1/cos(theta) + sin(theta)/cos(theta) - 1)
- Combine terms over a common denominator:
- Left Side = (sin theta) / ((1 + sin(theta) - cos(theta)) / cos(theta))
- Invert the denominator:
- Left Side = (sin theta * cos(theta)) / (1 + sin(theta) - cos(theta)).
Step 2: Simplifying the Right Side
- Start with the right side: 1 - (cos theta)/(cos cos theta - cot theta - 1).
- Recall that cot(theta) = cos(theta)/sin(theta).
- Substitute the definition:
- Right Side = 1 - (cos(theta)) / (cos^2(theta) - (cos(theta)/sin(theta)) - 1)
- Combine the terms in the denominator:
- Right Side = 1 - (cos(theta)) / ((cos^2(theta) - 1) - (cos(theta)/sin(theta)))
- Rewrite cos^2(theta) - 1 as -sin^2(theta):
- Right Side = 1 + (cos(theta))/(sin^2(theta) + (cos(theta)/sin(theta)))
- Simplifying gives:
- Right Side = (1 + sin(theta)) / (1 + sin(theta) - cos(theta)).
Conclusion
By simplifying both sides, we find that they are equal:
- Left Side = Right Side = (sin theta * cos(theta)) / (1 + sin(theta) - cos(theta)).
Thus, the identity is proved.