Recall the double angle identity for cosine:
[tex]\cos2\Phi=\cos^2\Phi-\sin^2\Phi[/tex]
So we have
[tex]\sin^2\Phi-\cos^2\Phi=-\cos2\Phi=0[/tex]
[tex]\cos x=0[/tex] for [tex]x=\dfrac{(2n+1)\pi}2[/tex], where [tex]n[/tex] is any integer, so
[tex]2\Phi=\dfrac{(2n+1)\pi}2\implies\Phi=\dfrac{(2n+1)\pi}4[/tex]
for any integer [tex]n[/tex]. We get solutions in the interval [tex]0\le\Phi<2\pi[/tex] when [tex]n=1,2,3,4[/tex], giving
[tex]\Phi=\dfrac\pi4,\dfrac{3\pi}4,\dfrac{5\pi}4,\dfrac{7\pi}4[/tex]
