How do you simplify the expression #cos(arctan(x/5)) #?

1 Answer
Nov 10, 2016

#cos(arctan(x/5))=+-root2(1/(1+(x/5)^2)#

Explanation:

From the fundamental identity of trigonometry #cos^2theta+sin^2theta=1#
we can deduce by dividing both sides for #cos^theta# and imposing the existence condition #theta ne pi/2+kpi#

#1+tan^2theta=1/cos^2theta# that can be rewritten as

#cos^2theta=1/(1+tan^2theta)# or

#costheta=+-root2(1/(1+tan^2theta)#
if #theta# is #arctan (x/5)# then
#cos(arctan(x/5))=+-root2(1/(1+tan^2(arctan(x/5))#
but #tan(arctan(x/5))=x/5#
in the end we can finally write
#cos(arctan(x/5))=+-root2(1/(1+(x/5)^2)#