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 the square of 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. Apply exponent rules where (e^u)^v=e^(u*v) and e^u⋅e^v=e^(u+v)

5. Simplify the terms:

(e^x)^2=e^(2*x)

−2*(e^x)*(e^(−x))=−2*e^(x−x)=−2*e^0=−2*(1)=−2

(e^(−x))^2=e^(−2*x)

6. Combine the simplified terms to reach the final expression.

Final Answer

(e^x−e^(−x))^2=e^(2*x)−2+e^(−2*x)

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