Write the given as two equations:

#cos(theta) = x" [1]"#

#theta=tan^-1(4/3)" [2]"#

and we want to find the value of x.

Use equation [2] to find the value of #tan(theta)# by applying the tangent function to both sides:

#tan(theta)=tan(tan^-1(4/3))" [2.1]"#

The tangent of its inverse reduces to #4/3# on the right:

#tan(theta) = 4/3" [2.2]"#

We can use the identity

#1 + tan^2(theta) = sec^2(theta)#

to give us a relationship between #tan(theta)# and #cos(theta)#:

#1 + (4/3)^2 = sec^2(theta)#

#9/9+16/9 = sec^2(theta)#

#25/9= sec^2(theta)#

We know that #sec^2(theta) = 1/cos^2(theta)#:

#25/9= 1/cos^2(theta)#

#cos^2(theta) = 9/25#

#cos(theta) = +-3/5#

We do not know whether we are in the 1st or 3rd quadrant, therefore, we must leave the #+-# as is.