BC

CD

ABCD

E

F

AEF

CEF

S_{\triangle ABE}=S_{1}

S_{\triangle ADF}=S_{2}

S_{1}+S_{2}

AE=AF=EF=a

\angle BAE=\alpha

S_{\triangle CEF}=S

\angle FAD=90^{\circ}-\alpha-60^{\circ}=30^{\circ}-\alpha,

\angle AFD=90^{\circ}-(30^{\circ}-\alpha)=60^{\circ}+\alpha,

\angle CFE=180^{\circ}-60^{\circ}-(60^{\circ}+\alpha)=60^{\circ}-\alpha,

S_{1}=\frac{1}{2}BE\cdot AE=\frac{1}{2}a\sin\alpha\cdot a\cos\alpha=\frac{1}{4}a^{2}\sin2\alpha,

S_{2}=\frac{1}{2}DF\cdot AD=\frac{1}{2}a\sin(30^{\circ}-\alpha)\cdot a\cos(30^{\circ}-\alpha)=\frac{1}{4}a^{2}\sin(60^{\circ}-2\alpha),

S=\frac{1}{2}EC\cdot CF=\frac{1}{2}a\sin(60^{\circ}-\alpha)\cdot a\cos(60^{\circ}-\alpha)=\frac{1}{4}a^{2}\sin(120^{\circ}-2\alpha),

\sin2\alpha+\sin(60^{\circ}-2\alpha)=2\sin\frac{2\alpha+(60^{\circ}-2\alpha)}{2}\cos\frac{2\alpha-(60^{\circ}-2\alpha)}{2}=

=2\sin30^{\circ}\cos(2\alpha-30^{\circ})=\cos(2\alpha-30^{\circ})=

=\sin(90^{\circ}-(2\alpha-30^{\circ}))=\sin(120^{\circ}-2\alpha),

S=S_{1}+S_{2}