Class 10 Exam > Class 10 Questions > Prove that sin square A by cos square A cos...
Start Learning for Free
Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt?
Most Upvoted Answer
Prove that sin square A by cos square A cos square A by sin squared ...
Proof of the Identity: sin²A/cos²A * cos²A/sin²A = 1/(sin²A*cos²A) - 2
----------------------------------------------------------------------------------
Proof using Trigonometric Identities:
1. Start with the left-hand side of the identity and simplify:
sin²A/cos²A * cos²A/sin²A = sin²A*sin²A / cos²A*cos²A
= (sin²A/cos²A)²
2. Use the Pythagorean Identity to rewrite sin²A/cos²A as tan²A + 1:
(sin²A/cos²A)² = (tan²A + 1)² / cos⁴A
= (tan⁴A + 2tan²A + 1) / cos⁴A
3. Use the Pythagorean Identity again to rewrite cos²A/sin²A as cot²A + 1:
(cos²A/sin²A)² = (cot²A + 1)² / sin⁴A
= (cot⁴A + 2cot²A + 1) / sin⁴A
4. Substitute these expressions back into the original identity:
(tan⁴A + 2tan²A + 1) / cos⁴A * (cot⁴A + 2cot²A + 1) / sin⁴A
= 1/(sin²A*cos²A) - 2
5. Simplify the expression on the right-hand side:
1/(sin²A*cos²A) - 2 = (1 - 2sin²A*cos²A) / (sin²A*cos²A)
= (1 - sin²2A) / (sin²A*cos²A)
= cos²2A / (sin²A*cos²A)
6. Substitute this expression back into the previous step:
(tan⁴A + 2tan²A + 1) / cos⁴A * (cot⁴A + 2cot²A + 1) / sin⁴A
= cos²2A / (sin²A*cos²A)
7. Simplify both sides of the equation:
tan⁴A + 2tan²A + 1 = cos²2A * sin⁴A / (cot⁴A + 2cot²A + 1)
8. Use the Pythagorean Identity to rewrite cot²A as 1 + tan²A:
tan⁴A + 2tan²A + 1 = cos²2A * sin⁴A / (1 + tan⁴A + 2tan²A)
9. Combine like terms:
tan⁴A - cos²2A * sin⁴A = -tan⁴A * cos²2A
10. Factor out tan²A:
tan²A (tan²A - cos²2A * sin²A) = -tan²A * cos²2A
11. Divide both sides by -tan²A:
tan²A - cos²2A * sin²A = -cos²2A
12. Use the double-angle identity for cosine:
tan²A - (1 - 2sin²A)sin²A = -1 + 2sin²A
----------------------------------------------------------------------------------
Proof using Trigonometric Identities:
1. Start with the left-hand side of the identity and simplify:
sin²A/cos²A * cos²A/sin²A = sin²A*sin²A / cos²A*cos²A
= (sin²A/cos²A)²
2. Use the Pythagorean Identity to rewrite sin²A/cos²A as tan²A + 1:
(sin²A/cos²A)² = (tan²A + 1)² / cos⁴A
= (tan⁴A + 2tan²A + 1) / cos⁴A
3. Use the Pythagorean Identity again to rewrite cos²A/sin²A as cot²A + 1:
(cos²A/sin²A)² = (cot²A + 1)² / sin⁴A
= (cot⁴A + 2cot²A + 1) / sin⁴A
4. Substitute these expressions back into the original identity:
(tan⁴A + 2tan²A + 1) / cos⁴A * (cot⁴A + 2cot²A + 1) / sin⁴A
= 1/(sin²A*cos²A) - 2
5. Simplify the expression on the right-hand side:
1/(sin²A*cos²A) - 2 = (1 - 2sin²A*cos²A) / (sin²A*cos²A)
= (1 - sin²2A) / (sin²A*cos²A)
= cos²2A / (sin²A*cos²A)
6. Substitute this expression back into the previous step:
(tan⁴A + 2tan²A + 1) / cos⁴A * (cot⁴A + 2cot²A + 1) / sin⁴A
= cos²2A / (sin²A*cos²A)
7. Simplify both sides of the equation:
tan⁴A + 2tan²A + 1 = cos²2A * sin⁴A / (cot⁴A + 2cot²A + 1)
8. Use the Pythagorean Identity to rewrite cot²A as 1 + tan²A:
tan⁴A + 2tan²A + 1 = cos²2A * sin⁴A / (1 + tan⁴A + 2tan²A)
9. Combine like terms:
tan⁴A - cos²2A * sin⁴A = -tan⁴A * cos²2A
10. Factor out tan²A:
tan²A (tan²A - cos²2A * sin²A) = -tan²A * cos²2A
11. Divide both sides by -tan²A:
tan²A - cos²2A * sin²A = -cos²2A
12. Use the double-angle identity for cosine:
tan²A - (1 - 2sin²A)sin²A = -1 + 2sin²A
Community Answer
Prove that sin square A by cos square A cos square A by sin squared ...
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
|
Explore Courses for Class 10 exam
|
|
Similar Class 10 Doubts
Top Courses for Class 10View all
Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt?
Question Description
Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? 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 that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt?.
Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? 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 that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt?.
Solutions for Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? defined & explained in the simplest way possible. Besides giving the explanation of
Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt?, a detailed solution for Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? has been provided alongside types of Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? theory, EduRev gives you an
ample number of questions to practice Prove that sin square A by cos square A cos square A by sin squared equal to 1 by sin square A into cos square A minus 2 Pls anyone ans fastly.its time 2 go 2 schl.pls fstttt? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup