It is given that x= tan hyperbolic inverse (1/2) which we can write as:

tanh(x) = 1/2. Now tanh(x)= sinh(x)/cosh(x) = (e^x-e^-x)/(e^x+e^-x) = (e^2x-1)/(e^2x+1)=1/2

Now using addendo-dividendo method we can say,

(e^2x-1+ e^2x+1)/(e^2x-1-e^2x-1)=(1+2)/(1-2) => e^2x=3 again we can say, e^-2x = 1/3.

Thus cosh(2x) = (e^2x+e^-2x)/2=5/3

The answer will be 5/3.

I hope this answer helps. All the very best for your future endeavors!

Hello candidate,

From the given value of tan inverse x we can get the value of tan x equal to 1/2.

Now, simplifying the expression for cos 2x we get its value is equal to 1 - 2 sin square x. From the value of tan x we can get the value of sin x equal to 1/√5. So, the value of sin^2x is equal to 1/5.

Now, the value of cos 2x is equal to 1- 2 sin square x= 1-1/5= 4/5.

Hope you found it helpful!!

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