Prove that:

$c o s 4 A = 1 - 8 c o s^{2} A + 8 c o s^{4} A$

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Solution

LHS = cos 4A

= cos 2.2A

$= 2 c o s^{2} 2 A - 1 [∵ c o s 2 θ = 2 c o s^{2} θ - 1] = 2 (2 c o s^{2} A - 1)^{2} - 1 = 2 (4 c o s^{4} A - 4 c o s^{2} A + 1) - 1 = 8 c o s^{4} A - 8 c o s^{2} A + 1 = 1 - 8 c o s^{2} A + 8 c o s^{4} A$

= RHS

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\cos 4 A = 1 - 8 \cos^{2} A + 8 \cos^{4} A

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