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Find the value of :- Sec^2x - cosec^2x / tan^2x - cot^2x , where pie € ( 0, pie/ 2 ) , (x is not equal to pie / 4 )? (€ refers to belongs to ) .?
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Find the value of :- Sec^2x - cosec^2x / tan^2x - cot^2x , where pie €...
Solution:
Given expression is Sec^2x - cosec^2x / tan^2x - cot^2x.
We know that,
Sec^2x - cosec^2x = (1/cos^2x) - (1/sin^2x) = (sin^2x - cos^2x) / (sin^2x cos^2x)
Also,
tan^2x - cot^2x = (sin^2x/cos^2x) - (cos^2x/sin^2x) = (sin^4x - cos^4x) / (sin^2x cos^2x)
Therefore,
Sec^2x - cosec^2x / tan^2x - cot^2x = [(sin^2x - cos^2x) / (sin^2x cos^2x)] / [(sin^4x - cos^4x) / (sin^2x cos^2x)]
= [(sin^2x - cos^2x) / (sin^4x - cos^4x)] * [sin^2x cos^2x / 1]
= [(sin^2x - cos^2x) / (sin^2x + cos^2x) (sin^2x - cos^2x)] * [sin^2x cos^2x]
= [(sin^2x cos^2x) / (sin^2x + cos^2x - 2cos^2x)]
= [(sin^2x cos^2x) / (sin^2x - cos^2x)]
= [(sin^2x cos^2x) / sin^2x] [1 / (1 - cos^2x / sin^2x)]
= cos^2x / (sin^2x - cos^2x)
As x is not equal to pi/4, we can substitute sin^2x = 1 - cos^2x.
Therefore,
cos^2x / (sin^2x - cos^2x) = cos^2x / (1 - 2cos^2x)
= 1 / (sec^2x - 2)
Hence, the value of the given expression is 1 / (sec^2x - 2) for x € (0, pi/2) and x is not equal to pi/4.
Given expression is Sec^2x - cosec^2x / tan^2x - cot^2x.
We know that,
Sec^2x - cosec^2x = (1/cos^2x) - (1/sin^2x) = (sin^2x - cos^2x) / (sin^2x cos^2x)
Also,
tan^2x - cot^2x = (sin^2x/cos^2x) - (cos^2x/sin^2x) = (sin^4x - cos^4x) / (sin^2x cos^2x)
Therefore,
Sec^2x - cosec^2x / tan^2x - cot^2x = [(sin^2x - cos^2x) / (sin^2x cos^2x)] / [(sin^4x - cos^4x) / (sin^2x cos^2x)]
= [(sin^2x - cos^2x) / (sin^4x - cos^4x)] * [sin^2x cos^2x / 1]
= [(sin^2x - cos^2x) / (sin^2x + cos^2x) (sin^2x - cos^2x)] * [sin^2x cos^2x]
= [(sin^2x cos^2x) / (sin^2x + cos^2x - 2cos^2x)]
= [(sin^2x cos^2x) / (sin^2x - cos^2x)]
= [(sin^2x cos^2x) / sin^2x] [1 / (1 - cos^2x / sin^2x)]
= cos^2x / (sin^2x - cos^2x)
As x is not equal to pi/4, we can substitute sin^2x = 1 - cos^2x.
Therefore,
cos^2x / (sin^2x - cos^2x) = cos^2x / (1 - 2cos^2x)
= 1 / (sec^2x - 2)
Hence, the value of the given expression is 1 / (sec^2x - 2) for x € (0, pi/2) and x is not equal to pi/4.
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Find the value of :- Sec^2x - cosec^2x / tan^2x - cot^2x , where pie € ( 0, pie/ 2 ) , (x is not equal to pie / 4 )? (€ refers to belongs to ) .?
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Find the value of :- Sec^2x - cosec^2x / tan^2x - cot^2x , where pie € ( 0, pie/ 2 ) , (x is not equal to pie / 4 )? (€ refers to belongs to ) .? 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 Find the value of :- Sec^2x - cosec^2x / tan^2x - cot^2x , where pie € ( 0, pie/ 2 ) , (x is not equal to pie / 4 )? (€ refers to belongs to ) .? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the value of :- Sec^2x - cosec^2x / tan^2x - cot^2x , where pie € ( 0, pie/ 2 ) , (x is not equal to pie / 4 )? (€ refers to belongs to ) .?.
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