How do you show #cos (arctan x) = [ 1 / ( sqrt(1 + x^2))]#?

1 Answer
Ratnaker Mehta
Jul 3, 2016

See the Proof given in the following Explanation.

Explanation:

Let #arctanx=theta rArr tantheta=x, x in RR, theta in (-pi/2,pi/2)#

Now, #sec^2theta=1+tan^2theta=1+x^2 rArr sectheta=+-sqrt(1+x^2)#

#:. costheta=+-1/sqrt(1+x^2)#

But, #theta in (-pi/2,pi/2) = Q_(IV) uu Q_I, where, costheta# is #+ve#.

Thus, #costheta=+1/sqrt(1+x^2)# and, replacing #theta# by #arctanx#, we have proved that,

#cos(arctanx)=1/sqrt(1+x^2).#

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