Given that \(\sin\) \(+ \cos\ \) \( = p\), verify that \(sin^4\ \) \(+ \cos^4\ \) \( =\) \(\frac{2 - (p^2 - 1)^2}{2}\).
Proof:
Given \(\sin\ \) \(+ \cos\ \) \(= p\)
Squaring on both sides.
\((\sin\ \) \(+ \cos\ \) \()^2 = p^2\)
\(\sin^2\ \) \(+ \cos^2\ \) \(+ 2\sin\ \) \(\cos\ \) \(= p^2\)
Hence proved.