Simplify and write the trigonometric expression in terms of sine and cosine:

((2+tan^2 x)(sec^2 x))-1 = (f(x))^2
f(x)= ???

To simplify and write the trigonometric expression in terms of sine and cosine, we need to replace the trigonometric functions tan and sec with their equivalent expressions in terms of sine and cosine.

Let's start by replacing tan^2 x with its equivalent expression in terms of sine and cosine:
tan^2 x = sin^2 x / cos^2 x

Similarly, we can replace sec^2 x with its equivalent expression:
sec^2 x = 1 / cos^2 x

Substituting these expressions into the initial expression:
((2 + (sin^2 x / cos^2 x))(1 / cos^2 x)) - 1 = (f(x))^2

Now, let's simplify this expression further by expanding and collecting like terms.

Expanding the expression:
((2cos^2 x + sin^2 x) / (cos^4 x)) - 1 = (f(x))^2

Combining the numerator:
(2cos^2 x + sin^2 x - cos^4 x) / (cos^4 x) = (f(x))^2

Now, let's combine like terms in the numerator:
(2cos^2 x - cos^4 x + sin^2 x) / (cos^4 x) = (f(x))^2

At this point, we have simplified the expression, but we can notice that the numerator can be factored. Let's factor the numerator:
(cos^2 x(2 - cos^2 x) + sin^2 x) / (cos^4 x) = (f(x))^2

Since (2 - cos^2 x) is equivalent to sin^2 x (using the Pythagorean identity: sin^2 x + cos^2 x = 1), we can simplify further:
((cos^2 x sin^2 x + sin^2 x) / (cos^4 x)) = (f(x))^2

Factoring out sin^2 x from the numerator:
(sin^2 x (1 + cos^2 x)) / (cos^4 x) = (f(x))^2

Using the Pythagorean identity sin^2 x = 1 - cos^2 x:
((1 - cos^2 x)(1 + cos^2 x)) / (cos^4 x) = (f(x))^2

Now, combine like terms in the numerator:
(1 - cos^4 x) / (cos^4 x) = (f(x))^2

At this point, we have the expression in terms of sine and cosine. The simplified and written form of the expression is:
f(x) = √(1 - cos^4 x) / cos^2 x