[tex]\displaystyle \sin^4 \alpha +\cos^4\alpha=(\sin^2\alpha)^2+(\cos^2\alpha)^2+2\times\sin^2\alpha\times\cos^2\alpha-\\ \\
-2\times\sin^2\alpha\times\cos^2\alpha=(\sin^2\alpha+\cos^2\alpha)^2-2\times\sin^2\alpha\times\cos^2\alpha=\\ \\
1-2\times(\sin\alpha\times\cos\alpha)^2\\ \\ \\
\sin(2\alpha)=\frac 12\Leftrightarrow 2\times\sin\alpha\times\cos\alpha=\frac 12\Leftrightarrow \sin\alpha\times\cos\alpha=\frac 14\\ \\ \\
1-2\times(\frac 14)^2=1-2\times \frac 1{16}=1-\frac 18=\frac 78\\ \\ \\ \\
c.t.d.[/tex]
