Question

find the simplified form of cos⁴ theta - sin 4 theta in term of sin theta​

Answer 1

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Answer 2

The simplified form of cos⁴θ - sin 4θ in terms of sinθ is:

-cos⁴θ + 6cos²θ - 4 = -[(1 - sin²θ)²] + 6(1 - sin²θ) - 4

= -1 + 8sin²θ - sin⁴θ

We can use the identity cos²θ + sin²θ = 1 to express cos⁴θ in terms of sin²θ:

cos⁴θ = (cos²θ)² = (1 - sin²θ)² = 1 - 2sin²θ + sin⁴θ

Similarly, we can use the identity sin²θ = 1 - cos²θ to express sin⁴θ in terms of cos²θ:

sin⁴θ = (sin²θ)² = (1 - cos²θ)² = 1 - 2cos²θ + cos⁴θ

Substituting these expressions into cos⁴θ - sin 4θ, we get:

cos⁴θ - sin 4θ = (1 - 2sin²θ + sin⁴θ) - sin(2θ)²

= 1 - 2sin²θ + sin⁴θ - 4sin²θ

= sin⁴θ - 6sin²θ + 1

Finally, substituting sin²θ = 1 - cos²θ, we get:

cos⁴θ - sin 4θ = (1 - cos⁴θ) - 6(1 - cos²θ) + 1

= -cos⁴θ + 6cos²θ - 4

The simplified form of cos⁴θ - sin 4θ in terms of sinθ is:

-cos⁴θ + 6cos²θ - 4 = -[(1 - sin²θ)²] + 6(1 - sin²θ) - 4

= -1 + 8sin²θ - sin⁴θ

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