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Prove that sin to the power 4A + sin square a by cos square a + cos square equal to 10 square a?
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Prove that sin to the power 4A + sin square a by cos square a + cos sq...
Understanding the Equation
To prove the identity sin^4(A) + sin^2(A)/cos^2(A) + cos^2(A) = 10sin^2(A), we will simplify the left-hand side step by step.
Step 1: Break Down sin^4(A)
- We can express sin^4(A) as (sin^2(A))^2.
- This allows us to rewrite the equation as (sin^2(A))^2 + sin^2(A)/cos^2(A) + cos^2(A).
Step 2: Use the Pythagorean Identity
- Recall that sin^2(A) + cos^2(A) = 1.
- Substitute cos^2(A) = 1 - sin^2(A) into the equation.
Step 3: Simplify the Equation
- The left side now becomes:
(sin^2(A))^2 + sin^2(A)/(1 - sin^2(A)) + (1 - sin^2(A)).
Step 4: Combine Like Terms
- Combine (1 - sin^2(A)) with (sin^2(A))^2 and the fraction.
- This results in a polynomial expression in terms of sin^2(A).
Step 5: Factor and Rearrange
- Factor the resulting polynomial if possible or rearrange it to match the right-hand side.
Step 6: Show Equality
- The right-hand side is 10sin^2(A).
- After simplification, show that both sides equal, thus proving the equation.
Conclusion
By following these steps, we demonstrate that sin^4(A) + sin^2(A)/cos^2(A) + cos^2(A) simplifies to 10sin^2(A), validating the identity.
To prove the identity sin^4(A) + sin^2(A)/cos^2(A) + cos^2(A) = 10sin^2(A), we will simplify the left-hand side step by step.
Step 1: Break Down sin^4(A)
- We can express sin^4(A) as (sin^2(A))^2.
- This allows us to rewrite the equation as (sin^2(A))^2 + sin^2(A)/cos^2(A) + cos^2(A).
Step 2: Use the Pythagorean Identity
- Recall that sin^2(A) + cos^2(A) = 1.
- Substitute cos^2(A) = 1 - sin^2(A) into the equation.
Step 3: Simplify the Equation
- The left side now becomes:
(sin^2(A))^2 + sin^2(A)/(1 - sin^2(A)) + (1 - sin^2(A)).
Step 4: Combine Like Terms
- Combine (1 - sin^2(A)) with (sin^2(A))^2 and the fraction.
- This results in a polynomial expression in terms of sin^2(A).
Step 5: Factor and Rearrange
- Factor the resulting polynomial if possible or rearrange it to match the right-hand side.
Step 6: Show Equality
- The right-hand side is 10sin^2(A).
- After simplification, show that both sides equal, thus proving the equation.
Conclusion
By following these steps, we demonstrate that sin^4(A) + sin^2(A)/cos^2(A) + cos^2(A) simplifies to 10sin^2(A), validating the identity.
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Prove that sin to the power 4A + sin square a by cos square a + cos square equal to 10 square a?
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Prove that sin to the power 4A + sin square a by cos square a + cos square equal to 10 square a? 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 to the power 4A + sin square a by cos square a + cos square equal to 10 square a? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that sin to the power 4A + sin square a by cos square a + cos square equal to 10 square a?.
Prove that sin to the power 4A + sin square a by cos square a + cos square equal to 10 square a? 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 to the power 4A + sin square a by cos square a + cos square equal to 10 square a? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that sin to the power 4A + sin square a by cos square a + cos square equal to 10 square a?.
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