Prove the identities:

1.)

2.)

thanks ahead :)

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- January 28th 2007, 01:44 AM^_^Engineer_Adam^_^proving hyperbolic Identities
Prove the identities:

1.)

2.)

thanks ahead :) - January 28th 2007, 02:09 AMticbol
1.) sinh(x+y) = sinh(x)cosh(y) +cosh(x)sin(y)

I have done this before.

I don't know how link, so here is a "copy-paste" of it:

----sinh u = (1/2)[e^u -e^(-u)]

----cosh u = (1/2)[e^u +e^(-u)]

----a^b * a^c = a^(b+c)

sinh(x+y) = sinh x cosh y + cosh x sinh y

We develop the RHS.

RHS =

= {(1/2)[e^x -e^(-x)] *(1/2)[e^y +e^(-y)]} +{(1/2)[e^x +e^(-x)] *(1/2)[e^y -e^(-y)]}

= (1/4)[e^x -e^(-x)][e^y +e^(-y)] +(1/4)[e^x +e^(-x)][e^y -e^(-y)]

Doing FOILs in the expansion,

= (1/4)[e^(x+y) +e^(x-y) -e^(-x+y) -e^(-x-y)] +(1/4)[e^(x+y) -e^(x-y) +e^(-x+y) -e^(-x-y)]

= (1/4)[e^(x+y) +e^(x-y) -e^(-x+y) -e^(-x-y) +e^(x+y) -e^(x-y) +e^(-x+y) -e^(-x-y)]

= (1/4)[2e^(x+y) -2e^(-x-y)]

= (1/2)[e^(x+y) -e^(-x-y)]

= (1/2)[e^(x+y) -e^[-(x+y)]

= sinh (x+y)

= LHS

Therefore, proven.

-------------------------------------------

2) tanh(ln(x)) = (x^2 -1) /(x^2 +1)

tanh(ln(x)) =

= sinh(ln(x)) /cosh(ln(x))

= (e^ln(x) -e^(-ln(x)) / (e^ln(x) +e^(-ln(x))

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

= (x -1/x) / (x +1/x)

= [(x^2 -1)/x] / [(x^2 +1)/x]

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

Proven.

Note:

e^(-ln(x))

= 1 /e^ln(x)

= 1 /x - January 28th 2007, 03:19 AM^_^Engineer_Adam^_^
- January 28th 2007, 06:13 AMThePerfectHacker
- January 28th 2007, 08:08 AMSoroban
Hello, ^_^Engineer_Adam^_^ !

This is ticbol's solution . . . in LaTeX.

. . (Don't mind me . . . I*love*doing this!)

Ticbol wrote:

. .

We have: .

We develop the RHS.

. .

Doing FOILs in the expansion:

. .

. .

. .

. .

Therefore, proven.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Multiply by

Therefore, proven.