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Prove the following identity
(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)?
(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)?
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Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) =...
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).
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).
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Question Description
Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)?.
Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cose theta- cot theta - 1)?.
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