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)= ???

( 2 + sin^2/cos^2)(1/cos^2) = 1 + y^2

(2/cos^2 + sin^2 /cos^4) = 1 + y^2

y^2 = (2 cos^2 + sin^2)/cos^4 - 1

y=+/- sqrt[ (2 cos^2 + sin^2)/cos^4 - 1 ]

that is not right according the program

To simplify and write the trigonometric expression in terms of sine and cosine, we will first rewrite the given trigonometric functions in terms of sine and cosine.

1. Start with the given expression: ((2 + tan^2 x)(sec^2 x)) - 1 = (f(x))^2

2. Recall the trigonometric identities:
- tan^2 x + 1 = sec^2 x (identity 1)
- sec x = 1/cos x (identity 2)

3. Substitute identity 1 into the expression:
((2 + tan^2 x)(sec^2 x)) - 1 becomes ((2 + sec^2 x) (sec^2 x)) - 1

4. Substitute identity 2 into the expression:
((2 + sec^2 x) (sec^2 x)) - 1 becomes ((2 + (1/cos^2 x)) (1/cos^2 x)) - 1

Now, let's simplify this expression further:

5. Multiply the numerators and denominators separately:
((2*cos^2 x + 1) / (cos^2 x)) - 1

6. Distribute the denominator inside the numerator:
(2*cos^2 x/cos^2 x + 1/cos^2 x) - 1

7. Cancel out the cos^2 x terms in the numerator:
2 + 1/cos^2 x - 1

8. Simplify the expression:
2 + (1/cos^2 x) - 1 = 2 + (sec^2 x - 1) = 2 + tan^2 x

Therefore, the simplified expression is 2 + tan^2 x.

Now, since f(x) is defined as the square root of the given expression, we can write:

f(x) = √(2 + tan^2 x)

Note: There might be alternative ways to manipulate the given expression, but the steps outlined above should help you simplify and write it in terms of sine and cosine.

( 2 + sin^2/cos^2)(1/cos^2) = 1 + y^2

(2 cos^2 + sin^2) /cos^4 = 1 + y^2

y^2 = (2 cos^2 + sin^2)/cos^4 - 1

y=+/- sqrt[ (2 cos^2 + sin^2)/cos^4 - 1 ]

well you could write it this way

y = +/- sqrt[ ( cos^2 +1)/cos^4 - 1]