Real are in the final Champions League spot, now only one point above Villarreal - who had never won at the Bernabeu.
Considering Madrid also drew 2-2 at home to Numancia in the cup last week, can Zidane survive?
Costa was suspended because of a red card for celebrating too close to the fans in the previous game.
The misfiring champions were beaten by an 87th-minute strike from Pablo Fornals at the end of a superb break by the visitors as Real pressed for a victor.
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In La Liga, Javi Calleja's side are in sixth, just four points behind Madrid, but they will be without midfielder Bruno Soriano, defender Ruben Semedo and reserve goalkeeper Andres Fernandez on Saturday, while forwards Nicola Sansone and Roberto Soldado are doubtful.
Nacho insists that there is nothing wrong with Real Madrid's attitude after Los Blancos' poor form continued with a demoralising 1-0 defeat to Villarreal.
Beaten 1-0 by Villarreal and with their title hopes in total disarray on a rain swept Saturday, Zidane's men trudged off the sodden pitch with jeers ringing around the Santiago Bernabeu Stadium.
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Chidambaram and is facing a probe for his alleged role in facilitating the FIPB's clearance during his father's tenure. According to initial reports, the raids were conducted in relation to the Aircel-Maxis case.
"We have endured negative results recently, but we know that we can change this situation by winning again", Zidane said.
Villarreal have won their last two away league games, both by the same scoreline (1-0).
Florin Andone headed in a late consolation for the home side with three minutes left.
Real - who saw a Cristiano Ronaldo free-kick turned on to the bar - had 28 shots, more than any other side in a La Liga game this season.
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Nothing that we like but something that can always happen when you're human. "We made too many easy mistakes which you can't do at this level".
Carlos Bacca has been involved in six of Villarreal's last 10 goals in La Liga (three goals, three assists).
Real controlled much of the first half, and had the first sight of goal on 17 minutes when Marcelo lashed a strike from distance which Sergio Asenjo turned away with a strong fingertip.
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He used an argon beam machine, an instrument created to seal bleeding blood vessels, to sear his initials "SB" into their livers. He resigned from his job at the hospital in 2014 after another surgeon found "SB" branded on a failed donor liver.