Find the square of cosine function based on the identity 4cosx-sin x=0.

We'll apply one of the 3 forms of fundamental formula of
trigonometry:

(tan x)^2 + 1 = 1/(cos
x)^2

(cos x)^2 = 1/[(tan x)^2 + 1]
(1)

Now, we'll find out tan x from the given
constraint:

4cos x - sin x =
0

We'll isolate cos x to the left
side:

4 cos x = sin x

We'll
divide by cos x both sides to create tangent function:

sin
x/cos x = 4

tan x = 4
(2)

We'll substitute (2) in
(1):

(cos x)^2 = 1/[(4)^2 +
1]

(cos x)^2 = 1/(16 + 1)

(cos
x)^2 = 1/17

The requested square of cosine
is: (cos x)^2 = 1/17.