Hyperbolic functions

By using the definitions of cosh u and sinh u in terms of e^u and e^-u, show that

sinh 2 u ≡ 2 sinh u cosh u.

You are now given that C has exactly one point of inflection.

Use your answer to part (b) to determine the exact coordinates of this point of inflection.

Give your answer in a logarithmic form where appropriate.

form.

In this question you must show detailed reasoning.

Using the definitions of sinh x and cosh x in terms of exponentials, show that

sinh 2 x ≡ 2 sinh x cosh x.

Solve the equation 15 sinh x + 16 cosh x - 6 sinh 2 x = 20, giving all your answers in

logarithmic form.

You are given that f (x) = 4 sinh x + 3 cosh x.

Show that the curve y = f (x) has no turning points.

Determine the exact solution of the equation f (x) = 5.

In this question you must show detailed reasoning.

Using the definition of cosh x in terms of exponentials, show that cosh 2x ≡ 2cosh² x - 1.

Solve the equation cosh 2 x = 3 cosh x + 1, giving all your answers in exact logarithmic form.

Using exponentials, show that cosh 2u ≡ 2 sinh²u + 1.

By differentiating both sides of the identity in part (a) with respect to u, show that

sinh 2u ≡ 2 sinh u cosh u.