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