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

5. Simplify the terms:

(ex)2=e(2*x)

−2*(ex)*(e(−x))=−2*e(x−x)=−2*e0=−2*(1)=−2

(e(−x))2=e(−2*x)

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

Final Answer

(ex−e(−x))2=e(2*x)−2+e(−2*x)

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