Given the functions:

f(x) = cosx.

g(x) = x^2.

We need to determine the value of (a) such that f(g(a)) = 1

First, let us determine the function f(g(x)).

f(g(x)) = f( x^2)

= cos(x^2).

==> f(g(x)) = cos(x^2).

Now we will substitute with x = a.

==> f(g(a)) = cos(a^2)

But, given that f(g(a)) = 1

==> f(g(a)) = cos(a^2) = 1

Then a^2 = 0

**==> a= 0.**

**Then the value of a is 0.**

Now are given that f(x) = cos x and g(x) = x^2.

If f(g(a)) = 1

=> f( a^2) = 1

=> cos a^2 = 1

=> a^2 = arc cos 1

=> a^2 = 0

=> a = 0

**Therefore a is equal to zero.**

To determine the value of a, we'll have to determine first the composition of the given functions f and g.

f(g(x)) is the result of composing f and g:

(fog)(x) = f(g(x))

To determine the expression of the composed function, we'll substitute x by g(x) and we'll get:

f(g(x)) = cos g(x)

Now, we'll substitute g(x) by it's expression:

f(g(x)) = cos x^2

Since we know the expression of f(g(x)), we can determine f(g(a)):

f(g(a)) = cos a^2

But, from enunciation, f(g(a)) = 1, so:

cos a^2 = 1

a^2 = +/-arccos 1 + 2*k*pi

a^2 = 0 + 2*k*pi

**a = +/-sqrt 2kpi**

**or**

**a = 0**