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

Problem

(e^x+e^(−x))^2

Solution

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

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

3. Substitute the values into the formula:

(e^x)^2+2*(e^x)*(e^(−x))+(e^(−x))^2

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

e^(2*x)+2*e^(x−x)+e^(−2*x)

5. Evaluate the middle term using the property e^0=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

(e^x+e^(−x))^2=e^(2*x)+2+e^(−2*x)

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