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

Problem

(e^x+e^(−x))*(e^x−e^(−x))−(e^x−e^(−x))^2

Solution

1. Identify the difference of squares pattern in the first part of the expression, where (a+b)*(a−b)=a^2−b^2

2. Apply the formula to the first term (e^x+e^(−x))*(e^x−e^(−x)) to get (e^x)^2−(e^(−x))^2

3. Simplify the exponents using the rule (e^a)^b=e^(a*b) resulting in e^(2*x)−e^(−2*x)

4. Expand the second part of the expression (e^x−e^(−x))^2 using the square of a binomial formula (a−b)^2=a^2−2*a*b+b^2

5. Substitute the values into the expansion to get (e^x)^2−2*(e^x)*(e^(−x))+(e^(−x))^2 which simplifies to e^(2*x)−2+e^(−2*x) because e^x⋅e^(−x)=e^0=1

6. Combine the two parts of the expression, ensuring the subtraction is distributed across the expanded terms.

e^(2*x)−e^(−2*x)−(e^(2*x)−2+e^(−2*x))

7. Distribute the negative sign.

e^(2*x)−e^(−2*x)−e^(2*x)+2−e^(−2*x)

8. Cancel the e^(2*x) and −e^(2*x) terms and combine the remaining like terms.

−2*e^(−2*x)+2

Final Answer

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

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