There's nothing fake:
http://www.cnn.com/2017/01/19/politics/nancy-sinatra-donald-trump-my-way/
They report about her tweeting - not sure that's serious news, but it's not "fake".
Don't like the headline - "Nancy Sinatra not happy..."? The headline doesn't state that she's unhappy with Trump - it says "not happy" - which clearly she isn't/wasn't. Perhaps she's indifferent - but clearly she wasn't thrilled that Trump was using her dad's songs.
Many performers are honored when their works are used for events such as this (they're "happy") - that clearly wasn't the case for Nancy Sinatra (regarding the use of her dad's song.)