The correct answer is (C) sec2θ.
Proof:
We know that:
tan2θ = sin2θ/cos2θ
Adding 1 to both sides, we get:
1 + tan2θ = 1 + sin2θ/cos2θ
Taking the common denominator, we get:
(1 * cos2θ + sin2θ)/cos2θ
Using the identity sin2θ + cos2θ = 1, we can simplify the numerator to:
cos2θ + sin2θ = 1
Substituting this back into the expression, we get:
(1 * cos2θ + sin2θ)/cos2θ = (1/1) / cos2θ
Recall that secθ = 1/cosθ, so we can write this as:
1/(cos2θ) = sec2θ
Therefore, 1 + tan2θ = sec2θ, and the correct answer is (C).