Trans alkenes are converted into their cis isomers and vice versa on epoxidation followed by treatment of the epoxide with triphenylphosphine.Propose a mechanism for the $\mathbf{epoxide}\to \mathbf{alkene}$reaction.

The epoxide is a three-member cyclic ring containing an oxygen atom.

So we can say that epoxide contains an oxygen atom and two carbon atoms in the ring structure. As the three-member ring is highly strained so it is very reactive.

When alkene is reacted with peracid they give epoxide. Trans alkene gives trans epoxide while cis alkene gives cis epoxide.

In this reaction, lone pair of phosphorus in triphenylphosphine is attacked on the carbon from the backside and the epoxide ring is open.

Attack on three membered ring

In the next step, the bond is rotated so that the stereochemistry of the group changes.

Bond Rotation

In the further step bond between oxygen and phosphorus form and we get inverted alkene it means in the given reaction trans alkene gives cis alkene.

Formation of Alkene