How do you evaluate csc(arctan(πx))?

1 Answer
Douglas K.
May 7, 2018

This webpage Inverse trigonometric functions contains a table entitled "Relationships between trigonometric functions and inverse trigonometric functions". In the table is the identity:

#sin(arctan(x)) = x/sqrt(x^2+1)#

We know that #csc(A) = 1/sin(A)# this tells us that we can invert the above equation:

#csc(arctan(x)) = sqrt(x^2+1)/x#

Substitute #x = pix#:

#csc(arctan(pix)) = sqrt((pix)^2+1)/(pix)#

Bring #1/pi # inside the radical as #1/pi^2#:

#csc(arctan(pix)) = sqrt(x^2+1/pi^2)/x#

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