[(1+cos(π)/(8)+i sin(π)/(8))/(1+cos(π)/(8)-i sin(π)/(8))]^8 is equal to (a) 1+i (b) 1-i (c) 1 (d

step 1 In the given question, the given expression is 1 plus cos pi by 8 plus i. sine pi by 8 upon 1 pl

[(1+cos(π)/(8)+i sin(π)/(8))/(1+cos(π)/(8)-i sin(π)/(8))]^8...

Key Concepts

Complex Numbers in Polar Form

This concept involves representing a complex number in the form r*(cos? + i*sin?), where r is the modulus and ? is the argument. It is a fundamental method for simplifying multiplication, division, and exponentiation of complex numbers by working with their magnitudes and angles instead of their Cartesian coordinates.

Euler's Formula

Euler's formula, expressed as e^(i?) = cos? + i*sin?, creates a bridge between exponential and trigonometric functions. This relationship is instrumental in converting between exponential and trigonometric forms of complex numbers, thereby simplifying calculations, especially when dealing with powers and roots of complex numbers.

De Moivre's Theorem

De Moivre's theorem states that (cos? + i*sin?)^n = cos(n?) + i*sin(n?) for any integer n. This theorem is essential when raising complex numbers to a power since it allows for the straightforward computation of the resulting modulus and argument by multiplying the angle by n.

Properties of Complex Conjugates

Complex conjugates are pairs of complex numbers that have identical real parts but opposite imaginary parts. Recognizing these pairs simplifies expressions, particularly when they appear in fractions, because the product of a complex number and its conjugate results in a real number, and operations involving them often lead to cancellations or simplifications in the expression.

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