Which of the following is a correct algebraic manipulation showing 1 + tan^2 θ equals sec^2 θ using sin and cos?

• A. 1 + tan^2 θ = (cos^2 θ + sin^2 θ)/cos^2 θ
• B. 1 + tan^2 θ = (cos^2 θ - sin^2 θ)/cos^2 θ
• C. 1 + tan^2 θ = (sin^2 θ + cos^2 θ)/sin^2 θ
• D. 1 + tan^2 θ = cos^2 θ

Answer: (A)

The main idea is to rewrite tan^2 θ in terms of sine and cosine and then use sin^2 θ + cos^2 θ = 1. Replace tan^2 θ with sin^2 θ / cos^2 θ:

1 + tan^2 θ = 1 + sin^2 θ / cos^2 θ.

Put over a common denominator cos^2 θ:

= (cos^2 θ)/cos^2 θ + sin^2 θ / cos^2 θ = (cos^2 θ + sin^2 θ) / cos^2 θ.

Since cos^2 θ + sin^2 θ = 1, this becomes 1 / cos^2 θ, which is sec^2 θ.

So this algebraic path shows exactly why 1 + tan^2 θ equals sec^2 θ. Other forms would lead to different expressions, such as cos 2θ over cos^2 θ, or 1 / sin^2 θ, or simply cos^2 θ, none of which equal sec^2 θ in general.