Show that \(\cos^4 x \equiv 1 - 2 \sin^2 x + \sin^4 x\).
Solution
Start with the identity \(\cos^2 x = 1 - \sin^2 x\).
Then, \(\cos^4 x = (\cos^2 x)^2 = (1 - \sin^2 x)^2\).
Expand \((1 - \sin^2 x)^2\):
\((1 - \sin^2 x)^2 = 1 - 2\sin^2 x + \sin^4 x\).
Thus, \(\cos^4 x = 1 - 2\sin^2 x + \sin^4 x\).
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