If \(\sin x = \cos x\), find \(\sin^4 x + \cos^4 x\).
\(\dfrac{1}{2}\)
\(\sin x = \cos x\) implies \(\tan x = 1\), so \(x = 45^\circ\) (in the principal range).
At \(x = 45^\circ\), \(\sin x = \cos x = \dfrac{1}{\sqrt{2}}\), hence \(\sin^2 x = \cos^2 x = \dfrac{1}{2}\).
\(\sin^4 x + \cos^4 x = \left(\dfrac{1}{2}\right)^2 + \left(\dfrac{1}{2}\right)^2 = \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{1}{2}\).
Hence the value is \(\dfrac{1}{2}\).
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