Without direct calculation of the Wronskian.

Question

Show that W(cos x,sin x, cosh(2x), cosh² x, 3) = 0 for all x,

Answer 1

If the Wronskian is zero, one of the functions is a linear combination of the others, so that is what we must show.

Consider the definition of Cosh(x), and what happens when you square it:

Cosh(x)= (e^x + e^-x )/2

Cosh²(x) = [(e^x + e^-x )/2]²

=(e^2x +2 + e^-2x)/4

=(1/2)(e^2x+e^-2x)/2 + 1/2

=(1/2)cosh(2x)+(1/6)(3)

Therefore, cosh²(x) is a linear combination of cosh(2x) and 3, so the wronskian of any set of functions containing these three functions will be zero for all x.