Prove that: 4x=1-8 ^2x+8 ^4x - Vidyadip

Question

Prove that:
$\cos4\text{x}=1-8\cos^2\text{x}+8\cos^4\text{x}$

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Answer

$\text{LHS}=\cos4\text{x}$
$=\cos2.2\text{x}$
$=2\cos^22\text{x}=1$ $[\because\cos2\theta=1\cos^2\theta-1]$
$=2(2\cos2\text{x}-1)^2-1$
$=2(4\cos^4\text{x}-4\cos^2\text{x}+1)-1$
$8\cos^4\text{x}-8\cos^4\text{x}+1$
$=1=8\cos^2\text{x}+8\cos^4\text{x}=\text{RHS}$

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