If x + iy = tan(A + iB), then what is the value of x2 + y2 + 2x cot 2A - 2?
2021
If x + iy = tan(A + iB), then what is the value of x2 + y2 + 2x cot 2A - 2?
- A.
1
- B.
-1
- C.
0
- D.
2
Attempted by 9 students.
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Correct answer: B
Given x + iy = tan(A + iB).
First, find x^2 + y^2. Since |tan(A + iB)|^2 = x^2 + y^2, we use the identity:
x^2 + y^2 = (cosh 2B - cos 2A) / (cosh 2B + cos 2A).
Next, find the real part x:
x = sin 2A / (cosh 2B + cos 2A).
Substitute x and (x^2 + y^2) into the expression: (cosh 2B - cos 2A)/(cosh 2B + cos 2A) + 2x cot 2A - 2.
Simplify the middle term: 2 * [sin 2A / (cosh 2B + cos 2A)] * [cos 2A / sin 2A] = 2 cos 2A / (cosh 2B + cos 2A).
Combine the fractions: [(cosh 2B - cos 2A) + 2 cos 2A] / (cosh 2B + cos 2A) - 2.
= (cosh 2B + cos 2A) / (cosh 2B + cos 2A) - 2 = 1 - 2 = -1.