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Prove that cosh 2x = 2 cosh^{2} x - 1

My working:

cosh x = 1/2(e^{x} + e^{-x})

cosh^{2} x = 1/4(e^{2x} + 2e^{0} + e^{-2x})

2cosh^{2} x = 1/2(e^{2x} + e^{-2x} + 2)

2cosh^{2} x = 1/2(e^{2x} + e^{-2x}) + 1

2cosh^{2} x - 1 = 1/2(e^{2x} + e^{-2x})

= cosh 2x

Is this enough? Are there any other ways to prove it?

My working:

cosh x = 1/2(e

cosh

2cosh

2cosh

2cosh

= cosh 2x

Is this enough? Are there any other ways to prove it?

You could note that coshx = cos(ix), and thus the identity cos2a = 2cos^{2}a - 1 holds.

Note though that sinhx = -isin(ix), and so cosh^{2}x = cos^{2}(ix) = 1 - sin^{2}(ix) = 1 + (-i)^{2}sin^{2}(ix) = 1 + sinh^{2}x. So it follows that cosh^{2}x - sinh^{2}x = 1, and also that cosh2x = cos2(ix) = cos^{2}(ix) - sin^{2}(ix) = cosh^{2}x + sinh^{2}x.

This can be verified using your method, of course:

We know that coshx = 1/2(e^{x} + e^{-x}), so cosh2x = 1/2(e^{2x} + e^{-2x}).

cosh^{2}x = 1/4(e^{2x} + e^{-2x} + 2)

sinh^{2}x = 1/4(e^{2x} + e^{-2x} - 2)

Adding,

cosh^{2}x + sinh^{2}x = 1/2(e^{2x} + e^{-2x}) = cosh2x.

Bit irrelevant, but meh, useful.

Note though that sinhx = -isin(ix), and so cosh

This can be verified using your method, of course:

We know that coshx = 1/2(e

cosh

sinh

Adding,

cosh

Bit irrelevant, but meh, useful.

Juwel

Looks like you did alright mate. I might have started with cosh(2x) = [e^(2x) + e^(-2x)]/2 and kept working.

I tried this way but couldn't get actually get anywhere. Any ideas on how to start it off?

Cheers guys for your help.

CalculusMan

I tried this way but couldn't get actually get anywhere. Any ideas on how to start it off?

Cheers guys for your help.

Cheers guys for your help.

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

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

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

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

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

= cosh^2 x - 1.

You've just proved something which isn't true.

I've followed your working but can't seem to spot how it should be (2cosh^x - 1)

I've followed your working but can't seem to spot how it should be (2cosh^x - 1)

Oops. Looks like I've forgotten how to think. Thanks.

Senh^2 (x) -2cosh^2(2)=1

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