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If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals to?
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If tan square A equals to 1 + 2 tan square B then cos square A upon co...
Problem Statement
Given the equation:
tan²A = 1 + 2tan²B
We need to find the value of cos²A / cos²B.
Step 1: Utilize the Identity
Recall the trigonometric identity:
tan²θ + 1 = sec²θ
This can be rearranged as:
sec²A = tan²A + 1 and sec²B = tan²B + 1
Step 2: Substitute the Values
From the given equation, we substitute for tan²A:
tan²A = 1 + 2tan²B
Thus,
sec²A = tan²A + 1 = (1 + 2tan²B) + 1 = 2 + 2tan²B
Step 3: Express Cosine in Terms of Tangent
Using the identity sec²θ = 1/cos²θ, we have:
cos²A = 1 / sec²A = 1 / (2 + 2tan²B) = 1 / (2(1 + tan²B))
For cos²B, we have:
cos²B = 1 / sec²B = 1 / (1 + tan²B)
Step 4: Find the Ratio
Now we can find the ratio of cos²A to cos²B:
cos²A / cos²B = (1 / (2(1 + tan²B))) / (1 / (1 + tan²B))
This simplifies to:
cos²A / cos²B = 1 / 2
Conclusion
Therefore, the final result is:
cos²A / cos²B = 1 / 2.
Given the equation:
tan²A = 1 + 2tan²B
We need to find the value of cos²A / cos²B.
Step 1: Utilize the Identity
Recall the trigonometric identity:
tan²θ + 1 = sec²θ
This can be rearranged as:
sec²A = tan²A + 1 and sec²B = tan²B + 1
Step 2: Substitute the Values
From the given equation, we substitute for tan²A:
tan²A = 1 + 2tan²B
Thus,
sec²A = tan²A + 1 = (1 + 2tan²B) + 1 = 2 + 2tan²B
Step 3: Express Cosine in Terms of Tangent
Using the identity sec²θ = 1/cos²θ, we have:
cos²A = 1 / sec²A = 1 / (2 + 2tan²B) = 1 / (2(1 + tan²B))
For cos²B, we have:
cos²B = 1 / sec²B = 1 / (1 + tan²B)
Step 4: Find the Ratio
Now we can find the ratio of cos²A to cos²B:
cos²A / cos²B = (1 / (2(1 + tan²B))) / (1 / (1 + tan²B))
This simplifies to:
cos²A / cos²B = 1 / 2
Conclusion
Therefore, the final result is:
cos²A / cos²B = 1 / 2.
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If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals to?
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If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals 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 If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals to? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals to?.
If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals 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 If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals to? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If tan square A equals to 1 + 2 tan square B then cos square A upon cos square B is equals to?.
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