It is said that when Ravana abducted Mother Sita, and Lord Ram was roaming around looking for Her, He was crying, shouting and moaning as He was so sad to have lost Her.
Of course, the Lord was playing a role on earth as a human, so He had to display human emotions.
Lord Shiva and Mother Sati came across this scene. Lord Shiva from far bowed down to His Lord.
Mother Sati said why are you bowing down?
Lord Shiva said because He is the Supreme Lord!
Mother Sati said that but He is crying like humans, how can He be God? I want to test Him.
So Lord Shiva said go ahead and test Him, but stay within your principles.
But Mother Sati changed Her form into the form of Mother Sita and went in front of Lord Ram, trying to confuse and fool Him.
Of course, Lord Ram knew she was Mother Sati and greeted Her and asked Her where is Lord Shiva?
She was ashamed and closed Her eyes. But even as Her eyes were closed, Lord Ram gave Her His Darshan along with Mother Sita in Her mind.
She went back to Lord Shiva but did not tell Him how She had tested Lord Ram.
But Lord Shiva knew!!!!!!!
And so He thought to Himself, that today She took the form of Mother Sita…………whom I worship! So from today I have to look at Sati as a Mother.
So He did not let Her sit next to Him anymore.
Sati could not bear it and always came in front of Lord Shiva with a crying face.
He got tired of seeing this and went into meditation for 87000 years!!!
And then, as we know, Sati went to the house of Her father, Daksh Prajapati, to attend a Yajna even though She and Lord Shiva were not invited.
When She heard Her Lord Shiva being insulted in that Yajna assembly, She could not take it and jumped into the fire and gave up Her life.
Moral of the story:
So the real reason that Lord Shiva let this happen, because She had not recognized Lord Ram, She had not listened to Him, and She had taken on the form of Mother Sita.
Of course, as we know, in a future birth Mother Sati did come back as Mother Parvati and attained Her beloved Lord Shiva after years of penance.
But let us keep in mind, that Sati did not actually doubt Lord Ram or Lord Shiva.
She just put on this act to prove to the world that Lord Ram was not a human because seeing how He went around crying for Mother Sita people might doubt His divinity.