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> Let $f(x)$ be a differentiable function in $[0,\frac{\pi }{2}]$. Prove that there is $0<c<\frac{\pi }{2}$ such that $$f'(c)=\left(f\left(\frac{\pi }{2}\right)-f(0)\right)\cos(c).$$
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$f(x)$ is a differentiable function in $[0,\frac{\pi }{2}]$ so it is also continuous in $[0,\frac{\pi }{2}]$. 

To prove this I need to use the mean value theorem.

Is this correct and how to continue from here?

thanks