Mathematics

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:
2 sec² θ - sec⁴ θ - 2 cosec² θ + cosec⁴ θ = cot⁴ θ - tan⁴ θ.

Trigonometric Identities

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Answer

The L.H.S of the equation can be written as,

⇒ 2(1 + tan² θ) - (sec² θ)² - 2(1 + cot² θ) + (cosec² θ)²
⇒ 2 + 2tan² θ - (1 + tan² θ)² - 2 - 2cot² θ + (1 + cot² θ)²
⇒ 2 + 2tan² θ - (1 + tan⁴ θ + 2tan² θ) - 2 - 2cot² θ + (1 + cot⁴ θ + 2cot² θ)
⇒ 2 + 2tan² θ - 1 - tan⁴ θ - 2tan² θ - 2 - 2cot² θ + 1 + cot⁴ θ + 2cot² θ
⇒ 2 - 2 + 2tan² θ - 2tan² θ + 2cot² θ - 2cot² θ + 1 - 1 + cot⁴ θ - tan⁴ θ
⇒ cot⁴ θ - tan⁴ θ.

Since, L.H.S. = R.H.S. hence, proved that 2 sec² θ - sec⁴ θ - 2 cosec² θ + cosec⁴ θ = cot⁴ θ - tan⁴ θ.

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Mathematics

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:
2 sec² θ - sec⁴ θ - 2 cosec² θ + cosec⁴ θ = cot⁴ θ - tan⁴ θ.

Trigonometric Identities

Answer

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