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What is sin4θ - cos4θ equal to for any real number θ?
- a)1
- b)1 - 2 sin2θ
- c)2 cos2θ + 1
- d)1 - 2 cos2θ
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
What is sin4θ - cos4θ equal to for any real number θ...
sin2 θ = 1 - cos2 θ
∴ sin4θ - cos4 θ = (1 - cos2θ)2 - cos4 θ
⇒ 1 - 2cos2 θ + cos4 θ - cos4 θ [∵ (a - b)2 = a2 - 2ab + b2]
⇒ 1 - 2cos2 θ
∴ sin4θ - cos4 θ = (1 - cos2θ)2 - cos4 θ
⇒ 1 - 2cos2 θ + cos4 θ - cos4 θ [∵ (a - b)2 = a2 - 2ab + b2]
⇒ 1 - 2cos2 θ
Most Upvoted Answer
What is sin4θ - cos4θ equal to for any real number θ...
Understanding the Expression sin4θ - cos4θ
To simplify sin4θ - cos4θ, we can utilize the identity for the difference of squares.
Step 1: Apply the Difference of Squares
- The expression can be rewritten as:
sin4θ - cos4θ = (sin2θ + cos2θ)(sin2θ - cos2θ)
Step 2: Recognize the Identity
- We know that sin2θ + cos2θ = 1 (Pythagorean identity).
- Therefore, the expression simplifies to:
(1)(sin2θ - cos2θ) = sin2θ - cos2θ
Step 3: Transforming sin2θ - cos2θ
- We can rewrite sin2θ as (1 - cos2θ) to facilitate further simplification:
sin2θ - cos2θ = (1 - cos2θ) - cos2θ = 1 - 2cos2θ
Conclusion
- Finally, we see that:
sin4θ - cos4θ = 1 - 2cos2θ
Thus, the correct answer is option 'D': 1 - 2 cos2θ.
This identity holds true for any real number θ and highlights a fundamental relationship between sine and cosine functions.
To simplify sin4θ - cos4θ, we can utilize the identity for the difference of squares.
Step 1: Apply the Difference of Squares
- The expression can be rewritten as:
sin4θ - cos4θ = (sin2θ + cos2θ)(sin2θ - cos2θ)
Step 2: Recognize the Identity
- We know that sin2θ + cos2θ = 1 (Pythagorean identity).
- Therefore, the expression simplifies to:
(1)(sin2θ - cos2θ) = sin2θ - cos2θ
Step 3: Transforming sin2θ - cos2θ
- We can rewrite sin2θ as (1 - cos2θ) to facilitate further simplification:
sin2θ - cos2θ = (1 - cos2θ) - cos2θ = 1 - 2cos2θ
Conclusion
- Finally, we see that:
sin4θ - cos4θ = 1 - 2cos2θ
Thus, the correct answer is option 'D': 1 - 2 cos2θ.
This identity holds true for any real number θ and highlights a fundamental relationship between sine and cosine functions.
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What is sin4θ - cos4θ equal to for any real number θ?a)1b)1 - 2 sin2θc)2 cos2θ + 1d)1 - 2 cos2θCorrect answer is option 'D'. Can you explain this answer?
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