How do you prove #sec^-1x+csc^-1x=pi/2#?

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Answer 1 · 2016-10-20T16:28:50

Please see the explanation.

Prove:

#sec^-1(x) + csc^-1(x) = pi/2#

Use the identity #csc^-1(x) = pi/2 - sec^-1(x)#:

#sec^-1(x) + pi/2 - sec^-1(x) = pi/2#

#pi/2 = pi/2#

Q.E.D.

Answer 2 · 2016-10-20T17:15:46

use the fact that csc is the complementary function of sec.

let
#sec^-1x=y#

#=>x=secy#

#=>x=csc(pi/2-y)#
#=>csc^-1x=pi/2-y#

substituting back for y

#csc^-1x=pi/2-sec^-1x#

hence #sec^-1x+csc^-1x=pi/2#

as required.