I was thinking about the problem that was as follows:
The integral equation $x(t)-\displaystyle \int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1,$ has
(a)no solution,
(b)a unique solution,
(c)more than one but finitely many solution,
(d)infinitely many solutions.
My attempts: From the given equation,we get $x(t)=\cos(t)\int_{0}^{1}\sec(s)x(s)ds + \sinh(t)=C \cos(t)+\sinh(t)$ where $C=\int_{0}^{1}\sec(s)x(s)ds$ and so $C=\int_{0}^{1}\sec(s)[\cos(s)C +\sinh(s)]ds=C\int_{0}^{1}ds+\int_{0}^{1}\sec(s)\sinh(s)ds$ and hence we get, $\int_{0}^{1}\sec(s)\sinh(s)ds=0.$
From here,i could not progress.Am i going in the right direction? Please help.Thanks in advance for your time.