24.98
24.99
24.100
24.101
24.102
24.103
24.104
24.105
24.106
24.107
24.108
206
CHAPTER 24
Show that cosh
2 x - sinh
2 x = 1.
This follows by direct computation from the definitions.
Let tanh x = sinh x/cosh x and sech x = 1 /cosh x. Find the derivative of tanh x.
Find D x (sech *).
Show that 1 - tanh2 * = sech2 x.
By Problem 24.98, cosh2 x - sinh2 * = 1. Dividing both sides by cosh2 x, we get 1 - tanh2 x = sech2 x.
In Problems 24.102-24.108, determine whether the given analogues of certain trigonometric identities also hold
for hyperbolic functions.
2 sinh x cosh x
The identity holds.
sinh (x + y) sinh x cosh y + cosh x sinh y.
sinh x cosh y + cosh x sinh y
The identity holds.
cosh (x + v) = cosh x cosh y - sinh x sinh y.
Think of y as fixed and take the derivatives of both sides of the identity in Problem 24.103. Then
cosh (x + y) = cosh x cosh y + sinh x sinh y. This is the correct identity, not the given one.
Thus, both identities hold.
By the identity found in the solution of Problem 24.104, cosh 2x = cosh
2 x + sinh
2 x. This is the correct
identity, not the given one.
By the identity established in the solution of Problem 24.106, cosh 2x = cosh
2 x + sinh
2 x. By the identity
cosh
2 x — sinh
2 x = \, sinh
2 x = cosh
2 x — \, and, therefore, cosh 2x = cosh
2 x + cosh
2 * — 1 = 2cosh
2 x - 1.
Thus, the identity is correct.
By the identity cosh2 x - sinh2 x = \, cosh2 x = sinh x + 1, and, substituting in the identity cosh2x =
cosh
2 x + sinh
2 x, we get cosh 2* = 1 + 2 sinh
2 x. This is the correct identity, not the given one.
cosh 2x ± 1 - 2 sinh
2 x.
cosh 2* =2= 2 cosh
2 x - 1.
cosh 2x — cosh
2 x - sinh
2 x.
cosh (-*) ^ cosh A: and sinh(-x) = -sinh (A:)
sinh 2^-2 sinh x cosh *.