Question

Prove that -
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Answer 1

Answer:

hey here is your proof

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Step-by-step explanation:

$to \: prove = \\ \\ 2.sec {}^{2} \: θ - sec {}^{4} \: θ- 2.cosec {}^{2} θ + cosec {}^{4} \: θ \\ \\ = \frac{1}{tan \: {}^{4} \:θ } - tan {}^{4} \: θ$

$LHS = 2sec {}^{2} \: θ - sec {}^{2} θ - 2.cosec {}^{2} θ + cosec {}^{4} \: θ \\ \\ we \: know \: that \\ 1 + tan {}^{2} \: θ = sec {}^{2} \: θ \\ \\ 1 + cot {}^{2} \: θ = cosec {}^{2} \: θ$

$= 2.(1 + tan {}^{2} \: θ) - (1 + tan {}^{2} θ) { }^{2} - 2.(1 + cot {}^{2} θ) + (1 + cot {}^{2} θ) {}^{2} \\ \\ = 2 + 2tan {}^{2} θ - (1 + tan {}^{4} θ + 2.tan {}^{2} θ) - 2 - 2.cot {}^{2} θ + (1 + cot {}^{4} θ + 2.cot {}^{2} θ) \\ \\ =2.tan {}^{2} θ - 1 - tan {}^{4} θ - 2.tan {}^{2} θ - 2cot {}^{2} θ + 1 + cot {}^{4} θ + 2.cot {}^{2} θ \\ \\ = - tan {}^{4} θ + cot {}^{4} θ \\ \\ = \frac{1}{tan {}^{4} θ} - tan {}^{4} θ \\ \\ = RHS \\ \\ thus \: proved$