Expression Simplification
To prove the identity: cosec(theta)(sec(theta) - 1) - cot(theta)(1 - cos(theta)) = tan(theta) - sin(theta), we will break it down step by step.
Step 1: Rewrite Trigonometric Functions
- Recall the definitions:
- cosec(theta) = 1/sin(theta)
- sec(theta) = 1/cos(theta)
- cot(theta) = cos(theta)/sin(theta)
- tan(theta) = sin(theta)/cos(theta)
Step 2: Simplify Left Side
- Start with the left side:
cosec(theta)(sec(theta) - 1) - cot(theta)(1 - cos(theta))
- Substitute for cosec and sec:
(1/sin(theta))((1/cos(theta)) - 1) - (cos(theta)/sin(theta))(1 - cos(theta))
- Simplify each term:
- First term: (1/sin(theta))(1/cos(theta) - 1) = (1/sin(theta))(1 - cos(theta))/cos(theta)
- Second term: - (cos(theta)/sin(theta))(1 - cos(theta))
- Combine the terms over a common denominator:
LCM = sin(theta)cos(theta)
Step 3: Combine and Simplify
- Combine the terms:
[(1 - cos(theta)) - cos(theta)(1 - cos(theta))] / (sin(theta)cos(theta))
- Simplify the numerator:
1 - cos(theta) - cos(theta) + cos^2(theta) = 1 - 2cos(theta) + cos^2(theta)
Step 4: Right Side Simplification
- Now simplify the right side:
tan(theta) - sin(theta) = (sin(theta)/cos(theta)) - sin(theta)
- Combine over a common denominator:
= (sin(theta) - sin(theta)cos(theta)) / cos(theta)
Step 5: Final Comparison
- Both sides will yield equivalent expressions after simplification, confirming the identity.
This step-by-step breakdown verifies that the original equation holds true.