Prove the following identities : sin4 A - cos4 A = 2sin2 A - 1

Solving L.H.S. of the equation : ⇒ (sin 2 A) 2 - (cos 2 A) 2 ⇒ (sin 2 A - cos 2 A)(sin 2 A + cos 2 A) By formula, sin 2 A + cos 2 A = 1 ⇒ (sin 2 A - cos 2 A) As, cos 2 A = 1 - sin 2 A ⇒ sin 2 A - (1 - sin 2 A) ⇒ sin 2 A - 1 + sin 2 A ⇒ 2sin 2 A - 1. Since, L.H.S. = R.H.S. Hence, proved that sin 4 A - cos 4 A = 2sin 2 A - 1.

Mathematics

Prove the following identities :
sin⁴ A - cos⁴ A = 2sin² A - 1

Trigonometric Identities

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Answer

Solving L.H.S. of the equation :

⇒ (sin² A)² - (cos² A)²

⇒ (sin² A - cos² A)(sin² A + cos² A)

By formula,

sin² A + cos² A = 1

⇒ (sin² A - cos² A)

As, cos² A = 1 - sin² A

⇒ sin² A - (1 - sin² A)

⇒ sin² A - 1 + sin² A

⇒ 2sin² A - 1.

Since, L.H.S. = R.H.S.

Hence, proved that sin⁴ A - cos⁴ A = 2sin² A - 1.

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Mathematics

Prove the following identities :sin4 A - cos4 A = 2sin2 A - 1

Trigonometric Identities

Answer

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Prove the following identities :
sin⁴ A - cos⁴ A = 2sin² A - 1

A conical tent is to accommodate 77 persons. Each person must have 16 m³ of air to breathe. Given the radius of the tent as 7 m, find the height of the tent and also its curved surface area.