Sivakumar
MATH 171
Exercise Set 7b
1. Prove the following identities:
(i) sinh(−x) = − sinh x, cosh(−x) = cosh x
(ii) cosh2 x − sinh2 x = 1
(iii) sinh(x + y) = sinh x cosh y + cosh x sinh y
(iv) cosh(x + y) = cosh x cosh y + sinh x sinh y
(v) (cosh x+sinh x)n = cosh(nx)+sinh(nx) for every real number x and every positive integer
n.
2. Let C denote the curve y = cosh(x), −∞ < x < ∞. Find the point on C where the slope of
the tangent is 1.
3. If x = ln(sec θ + tan θ), show that cosh x = sec θ.
The hyperbolic tangent function is defined as follows:
tanh x :=
ex − e−x
sinh x
= x
,
cosh x
e + e−x
−∞ < x < ∞.
This function has domain (−∞, ∞), and its range is the interval (−1, 1). For a graph of y = tanh x,
see page 283 in the text.
4. Show that
sin−1 (tanh x) = tan−1 (sinh x),
1
x ∈ R.