Simplify (e^x+e^(-x))^2

Problem

(ex+e(−x))2

Solution

1. Identify the expression as a binomial square of the form (a+b)2 where a=ex and b=e(−x)

2. Apply the formula for squaring a binomial: (a+b)2=a2+2*a*b+b2

3. Substitute the values into the formula:

(ex)2+2*(ex)*(e(−x))+(e(−x))2

4. Simplify the terms using exponent rules, specifically (eu)v=e(u*v) and eu⋅ev=e(u+v)

e(2*x)+2*e(x−x)+e(−2*x)

5. Evaluate the middle term using the property e0=1

e(2*x)+2*(1)+e(−2*x)

6. Combine the terms into the final simplified form:

e(2*x)+2+e(−2*x)

Final Answer

(ex+e(−x))2=e(2*x)+2+e(−2*x)

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