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Prove the following: 2 sec2θ – sec4θ – 2cosec2θ + cosec4θ = cot4θ – tan4θ - Mathematics and Statistics

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Question

Prove the following:

2 sec²θ – sec⁴θ – 2cosec²θ + cosec⁴θ = cot⁴θ – tan⁴θ

Sum

Solution

L.H.S. = 2 sec²θ – sec⁴θ – 2cosec²θ + cosec⁴θ

⁼2 sec²θ – (sec²θ)² – 2cosec²θ + (cosec²θ)²

= 2(1 + tan²θ) – (1 + tan²θ)² – 2(1 + cot²θ) + (1 + cot²θ)²

= 2 + 2tan²θ – (1 + 2tan²θ + tan⁴θ) – 2 – 2cot²θ + 1 + 2cot²θ + cot⁴θ

= 2 + 2tan²θ – 1 – 2tan²θ – tan⁴θ – 2 – 2cot²θ + 1 + 2cot²θ + cot⁴θ

= cot⁴θ – tan⁴θ

= R.H.S.

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Fundamental Identities

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Q 10) iv)Q 10) iii)Q 10) v)

Chapter 2: Trigonometry - 1 - MISCELLANEOUS EXERCISE - 2 [Page 33]

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Balbharati Mathematics and Statistics 1 (Arts and Science) [English] 11 Standard Maharashtra State Board

Chapter 2 Trigonometry - 1
MISCELLANEOUS EXERCISE - 2 | Q 10) iv) | Page 33

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