(1+i/1-i)^2+(1-i/1+i)^2 Is Equal To

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (1+i/1-i)^2+(1-i/1+i)^2

This problem involves simplifying a complex expression involving fractions and powers of complex numbers. Let's break it down step-by-step:

Simplifying the Fractions

1. Rationalizing the Denominators: To simplify the fractions, we'll multiply each fraction by the conjugate of its denominator. The conjugate of 1-i is 1+i, and the conjugate of 1+i is 1-i.

   • (1+i)/(1-i) * (1+i)/(1+i) = (1 + 2i + i^2) / (1 - i^2)
   • (1-i)/(1+i) * (1-i)/(1-i) = (1 - 2i + i^2) / (1 - i^2)

2. Simplifying using i^2 = -1: Remember that i^2 is equal to -1. Substituting this into our simplified fractions:

   • (1 + 2i - 1) / (1 + 1) = 2i / 2 = i
   • (1 - 2i - 1) / (1 + 1) = -2i / 2 = -i

Squaring the Simplified Expressions

Now, our expression becomes:

(1+i/1-i)^2+(1-i/1+i)^2 = (i)^2 + (-i)^2

Final Calculation

Finally, we can calculate the result:

• i^2 + (-i)^2 = -1 + (-1) = -2

Therefore, (1+i/1-i)^2+(1-i/1+i)^2 is equal to -2.