simplify sinh(ln(t)) by rewriting it into a rational expression.\nsinh(ln(t)) =

Explanation:

Step1: Recall the definition of hyperbolic sine

The definition of (\sinh(x)=\frac{e^{x}-e^{-x}}{2}). So, (\sinh(\ln(t))=\frac{e^{\ln(t)}-e^{-\ln(t)}}{2}).

Step2: Simplify the exponential - logarithmic expressions

We know that (e^{\ln(t)} = t) and (e^{-\ln(t)}=e^{\ln(t^{-1})}=t^{-1}=\frac{1}{t}) (using the properties (a\ln(b)=\ln(b^{a})) and (e^{\ln(u)} = u) for (u>0)). Substitute these into the expression: (\sinh(\ln(t))=\frac{t-\frac{1}{t}}{2}).

Step3: Simplify the rational expression

(\frac{t-\frac{1}{t}}{2}=\frac{\frac{t^{2}-1}{t}}{2}=\frac{t^{2}-1}{2t}) (by getting a common denominator in the numerator and then dividing by 2).

Answer:

(\frac{t^{2}-1}{2t})