I assume that you mean cos(arctan(3/4)), where arctan(x) is the inverse function of tan(x).
(Sometimes arctan(x) as written as tan^-1(x), but personally I find it confusing as it could be possibly misunderstood as 1/tan(x) instead.)
We need to use the following identities:
cos(x)=1/sec(x) {Identity 1}
tan^2(x)+1=sec^2(x), or sec(x)=sqrt(tan^2(x)+1) {Identity 2}
With these in mind, we can find cos(arctan(3/4)) easily.
\ \ \ \ \ \ \ cos(arctan(3/4))
=1/sec(arctan(3/4)) {Using Identity 1}
=1/sqrt(tan(arctan(3/4))^2+1) {Using Identity 2}
=1/sqrt((3/4)^2+1) {By definition of arctan(x)}
=4/5
