Prove that for x is greater than or equal to 1: arccos(1/x) = arcsin((sqrt(x^2 -1)/x) Anyone know how to do this?

Question

2223 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-26T20:24:30

Please see below.

Here,

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

Take,

#cos^-1(1/x)=theta=>1/x=costheta=>x=1/costheta=sectheta#

Given that,

#x>=1#

#=>x>0 andx>=1#

#=>1/x>0 and 1/x<=1#

i.e. #0 < 1/x <=1=>0 < costheta <=1=>0<= theta < pi/2#

Now,

#RHS=sin^-1((sqrt(x^2-1))/x),....#where, #x=sectheta#

#=sin^-1(sqrt(sec^2theta-1)/(sectheta))#

#=sin^-1(sqrt(tan^2theta)/sectheta)....#where, #sec^2theta-1=tan^2theta#

#=sin^-1(tantheta/sectheta)#

#=sin^-1((sintheta/cancel(costheta))/(1/cancelcostheta))#

#=sin^-1(sintheta).....#where, #0<= theta < pi/2#

#=theta#

#=cos^-1(1/x)#

#=LHS#