Choose the correct alternative answer for the following questions. 1 + tan2θ = ? (A) cot2θ (B) cosec2θ (C) sec2θ (D) tan2θ

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).