How do you simplify #cos(sin^-1 x)#?

1 Answer
Subha · Jim H
Oct 21, 2016

±#sqrt (1-x^2)#

Explanation:

#cos(sin^-1 x)#
Let,
#sin^-1x = theta#
#=>sin theta = x#
#=>sin^2theta =x^2#
#=>1-cos^2theta = x^2#
#=>cos^2theta = 1-x^2#
#=>cos theta =± sqrt (1-x^2) #
#=>theta =cos^-1±sqrt(1-x^2)#
Putting this,
#cos(cos^-1±sqrt(1-x^2))#
#=±sqrt(1-x^2)#

But #sin^(-1)x# is, by definition, in #[-pi/2,pi/2]# so #cos(sin^-1x) >= 0#

so #cos(sin^-1x) = sqrt(1-x^2)#

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