How to calculate the secant of a complex number?

To calculate the secant of a complex number, use the formula sec(z) = 1/cos(z).

The secant of a complex number z is defined as the reciprocal of the cosine of z. To calculate the cosine of a complex number, use the formula cos(z) = (e^(iz) + e^(-iz))/2.

Let z = x + yi, where x and y are real numbers. Then, e^(iz) = e^(ix-y) = e^(-y)e^(ix), and e^(-iz) = e^(y-xi) = e^(-y)e^(ix).

Substituting these expressions into the formula for cos(z), we get:

cos(z) = (e^(iz) + e^(-iz))/2
= (e^(-y)e^(ix) + e^(-y)e^(-ix))/2
= e^(-y)/2 (e^(ix) + e^(-ix))
= e^(-y)/2 (cos(x) + i sin(x) + cos(x) - i sin(x))
= e^(-y) cos(x)

Therefore, the secant of z is:

sec(z) = 1/cos(z)
= 1/(e^(-y) cos(x))
= e^y / cos(x)

So, the secant of a complex number z = x + yi is e^y / cos(x).

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