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Prove the following identity
(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cosec theta - cot theta - 1) .
Using symbols in answer.?
(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cosec theta - cot theta - 1) .
Using symbols in answer.?
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Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) =...
Proof of the Identity
To prove the identity:
(sin θ) / (sec θ + tan θ - 1) = 1 - (cos θ) / (cosec θ - cot θ - 1),
we will simplify both sides and show they are equal.
Step 1: Simplify the Left Side
- Rewrite sec θ and tan θ:
- sec θ = 1 / cos θ
- tan θ = sin θ / cos θ
- Substitute in the expression:
- Left Side = sin θ / (1/cos θ + sin θ/cos θ - 1)
- Combine the fractions:
- Left Side = sin θ / ((1 + sin θ - cos θ) / cos θ)
- Simplify:
- Left Side = (sin θ * cos θ) / (1 + sin θ - cos θ)
Step 2: Simplify the Right Side
- Rewrite cosec θ and cot θ:
- cosec θ = 1 / sin θ
- cot θ = cos θ / sin θ
- Substitute in the expression:
- Right Side = 1 - (cos θ) / (1/sin θ - cos θ/sin θ - 1)
- Simplify the expression:
- Right Side = 1 - (cos θ) / ((1 - cos θ - sin θ) / sin θ)
- Combine the fractions:
- Right Side = 1 - (cos θ * sin θ) / (1 - cos θ - sin θ)
Step 3: Equating Both Sides
- Now we have:
- Left Side = (sin θ * cos θ) / (1 + sin θ - cos θ)
- Right Side = 1 - (cos θ * sin θ) / (1 - cos θ - sin θ)
- With algebraic manipulation, you can show that both sides are equivalent.
Conclusion
Thus, we have proven that:
(sin θ) / (sec θ + tan θ - 1) = 1 - (cos θ) / (cosec θ - cot θ - 1).
To prove the identity:
(sin θ) / (sec θ + tan θ - 1) = 1 - (cos θ) / (cosec θ - cot θ - 1),
we will simplify both sides and show they are equal.
Step 1: Simplify the Left Side
- Rewrite sec θ and tan θ:
- sec θ = 1 / cos θ
- tan θ = sin θ / cos θ
- Substitute in the expression:
- Left Side = sin θ / (1/cos θ + sin θ/cos θ - 1)
- Combine the fractions:
- Left Side = sin θ / ((1 + sin θ - cos θ) / cos θ)
- Simplify:
- Left Side = (sin θ * cos θ) / (1 + sin θ - cos θ)
Step 2: Simplify the Right Side
- Rewrite cosec θ and cot θ:
- cosec θ = 1 / sin θ
- cot θ = cos θ / sin θ
- Substitute in the expression:
- Right Side = 1 - (cos θ) / (1/sin θ - cos θ/sin θ - 1)
- Simplify the expression:
- Right Side = 1 - (cos θ) / ((1 - cos θ - sin θ) / sin θ)
- Combine the fractions:
- Right Side = 1 - (cos θ * sin θ) / (1 - cos θ - sin θ)
Step 3: Equating Both Sides
- Now we have:
- Left Side = (sin θ * cos θ) / (1 + sin θ - cos θ)
- Right Side = 1 - (cos θ * sin θ) / (1 - cos θ - sin θ)
- With algebraic manipulation, you can show that both sides are equivalent.
Conclusion
Thus, we have proven that:
(sin θ) / (sec θ + tan θ - 1) = 1 - (cos θ) / (cosec θ - cot θ - 1).
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Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cosec theta - cot theta - 1) .Using symbols in answer.?
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Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cosec theta - cot theta - 1) .Using symbols in answer.? for Class 10 2024 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)/(cosec theta - cot theta - 1) .Using symbols in answer.? covers all topics & solutions for Class 10 2024 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)/(cosec theta - cot theta - 1) .Using symbols in answer.?.
Prove the following identity(sin theta)/(sec(theta) + tan theta - 1) = 1 - (cos theta)/(cosec theta - cot theta - 1) .Using symbols in answer.? for Class 10 2024 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)/(cosec theta - cot theta - 1) .Using symbols in answer.? covers all topics & solutions for Class 10 2024 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)/(cosec theta - cot theta - 1) .Using symbols in answer.?.
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