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

Problem

(ex+e(−x))*(ex−e(−x))−(ex−e(−x))2

Solution

1. Identify the difference of squares pattern in the first part of the expression, where (a+b)*(a−b)=a2−b2

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

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

4. Expand the second part of the expression (ex−e(−x))2 using the square of a binomial formula (a−b)2=a2−2*a*b+b2

5. Substitute the values into the expansion to get (ex)2−2*(ex)*(e(−x))+(e(−x))2 which simplifies to e(2*x)−2+e(−2*x) because ex⋅e(−x)=e0=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

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

Want more problems? Check here!