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

1 Answer
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).