If \(\sin^4 x - \cos^4 x = \frac{1}{5}\), then the value of \(2\sin^2 x - 1\) is:
\(\frac{1}{5}\)
Factorise as a difference of squares: \(\sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x)\).
Since \(\sin^2 x + \cos^2 x = 1\), this equals \(\sin^2 x - \cos^2 x\).
Using \(\cos^2 x = 1 - \sin^2 x\): \(\sin^2 x - (1 - \sin^2 x) = 2\sin^2 x - 1\).
So \(2\sin^2 x - 1 = \sin^4 x - \cos^4 x = \frac{1}{5}\).
Hence, the answer is \(\frac{1}{5}\).
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