OK so not having finished season two yet...
Claire obviously a strong woman but displays moments of weakness where Frank has not. That's not to say Frank doesn't HAVE weaknesses (Like the teacher's strike debate he royally ****) but that he as a person does not tolerate weakness. He's a man of action, of pragmatism.
Their relationship is one of Trust, but it's a game they play with each other. Claire new well that Frank was **** Zoe barnes, and as a viewer we knew that she and Frank were OK with this as a means to an end. A public mouthpiece they could control for their own gain. Frank trusts that Claire has their combined interests in mind at all times but when Claire feels like her own contributions are being made second to Franks, her weakness shows again as she makes a play through Remy to acquire the funds she needed for her mission, as well as when she deliberately sabotages Frank's bill votes by not persuading the two votes he thought he could count on.
She did this knowing that Frank would understand that she is his equal, his counter part, his female form. She is every bit as ruthless as he is - consider when she fires her number 1 employee and then cancels her health insurance for which her HMO won't provide the medicine she needs to keep her baby alive. She tells her "Don't think for one minute I wouldn't see that baby die inside you" - to accomplish her goals, and to that end, further Frank's.
They both crave ambition and power. They don't seem to care for money or fame, just power. They want to be at the top and they will destroy anyone they have to, to accomplish this. Their shared desire- lust even- for power is at the core of their relationship. They do what they can to help each other but they are both intensely selfish individuals. Frank's end game, one would think, is the Presidency. Claire's is yet to be revealed as far as I know, but she is happy to resign from her post if her former employee will drop the lawsuit, so she exists merely to prop Frank up and advise him.
Their relationship is business-formal, but really it's also one of deep trust - trust without question. I think this is why Claire undermines Frank in season 1, to re-affirm their trust, not just because she's felt like Frank is always ahead of her - he is, but because that is the core tenant of their relationship. She knows he is a killer, she knows he is ruthless. She loves him for it and know's "she'll never be bored" with a man like Frank. Claire is still the weaker character which is made evident by their focusing on her Photographer romance, as well as her menopause, and her need to show Frank that she's as ruthless as he is.
Well those are my thoughts, It'll be interesting to see how season two continues to develop those two, I'm on a few episodes in.
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