The Liberal Democrats, who are hoping to shred key elements of the NHS reforms, believe they showed their influence in government on Wednesday on the highly sensitive matter of the Trident nuclear deterrent.

Liam Fox, the defence secretary, told MPs that the Cabinet Office would carry out a study to review the "costs, feasibility and credibility" of an alternative nuclear deterrent.

The study, which will be overseen by the Lib Dem armed forces minister Nick Harvey, is in the spirit of the coalition agreement. This said that the Lib Dems would "continue to make the case for alternatives" to Trident.

But will the Lib Dem study make any difference? Not if Fox has anything to do with it.

The defence secretary, who regularly talks of his hope that the Tories will govern without the Lib Dems after the next election, had a broad grin on his face as he walked around the corridors of Westminster after making his statement on Wednesday afternoon. Fox let it be known that he thought the review was a joke and a complete waste of time that will do nothing to change Britain's "continuous at sea deterrent".

Fox is confident because he believes that the Lib Dems will never overcome a powerful political force that wants to see the retention the key principle behind Trident – that Britain can launch a nuclear attack from a submarine from anywhere in the world at any time. His friend, the former shadow defence minister Julian Lewis, outlined his thinking in this question to David Cameron shortly before Fox's statement:

Would he give his blessing to honourable members in the Conservative party and on the Labour benches who, like him, think that the nuclear deterrent should be above party politics, if they formed an alliance on this important issue, just as we did so successfully on the Alternative Vote?

The prime minister laughed as he gave this light-hearted response: