Prove the following identities : sec 4 A (1 - sin 4 A) - 2 tan 2 A = 1

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

Mathematics

Prove the following identities :
sec⁴ A (1 - sin⁴ A) - 2 tan² A = 1

Trigonometric Identities

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Answer

Solving L.H.S. of the above equation :

⇒ sec⁴ A (1 - sin⁴ A) - 2 tan² A

⇒ sec⁴ A (1 - sin² A)(1 + sin² A) - 2 tan² A

By formula,

1 - sin² A = cos² A

⇒ sec⁴ A cos² A (1 + sin² A) - 2 tan² A

⇒ sec⁴ A $\times \dfrac{1}{\text{sec}^2 A}$ (1 + sin² A) - 2 tan² A

⇒ sec² A (1 + sin² A) - 2 tan² A

⇒ sec² A + sec² A sin² A - 2 tan² A

⇒ sec² A + $\dfrac{1}{\text{cos}^2 A} \times$ sin² A - 2 tan² A

⇒ sec² A + tan² A - 2 tan² A

⇒ sec² A - tan² A

⇒ 1.

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

Hence, proved that sec⁴ A (1 - sin⁴ A) - 2 tan² A = 1.

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Mathematics

Prove the following identities :
sec⁴ A (1 - sin⁴ A) - 2 tan² A = 1

Trigonometric Identities

Answer

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