Question
Download Solution PDFIf cos4 θ - sin4 θ = 4/5, then find the value of sin 4θ.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
cos⁴θ - sin⁴θ = 4/5
Formula used:
1. Difference of squares identity: cos⁴θ - sin⁴θ = (cos²θ - sin²θ)(cos²θ + sin²θ)
2. cos²θ + sin²θ = 1
3. cos 2θ = cos²θ - sin²θ
4. sin 4θ = 2sin 2θ cos 2θ
Calculation:
cos⁴θ - sin⁴θ = (cos²θ - sin²θ)(cos²θ + sin²θ)
⇒ cos⁴θ - sin⁴θ = cos²θ - sin²θ (since cos²θ + sin²θ = 1)
So, cos²θ - sin²θ = 4/5
cos 2θ = cos²θ - sin²θ
⇒ cos 2θ = 4/5
Find sin 2θ using the Pythagorean identity
sin² 2θ + cos² 2θ = 1
Substitute cos 2θ = 4/5:
sin² 2θ + (4/5)² = 1
sin² 2θ + 16/25 = 1
sin² 2θ = 1 - 16/25 = 9/25
⇒ sin 2θ = 3/5
sin 4θ = 2sin 2θ cos 2θ
Substitute sin 2θ = 3/5 and cos 2θ = 4/5:
⇒ sin 4θ = 2 × (3/5) × (4/5) = 24/25
∴ The value of sin 4θ is 24/25.
Last updated on Jun 26, 2025
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