For each expression in Column I, choose the expression from Column II that completes an identity.
6. sec² x = ____

Verified step by step guidance

1. Recall the Pythagorean identity involving secant and tangent: \(\sec^2 x = 1 + \tan^2 x\).
2. Understand that this identity comes from dividing the fundamental identity \(\sin^2 x + \cos^2 x = 1\) by \(\cos^2 x\).
3. Rewrite the expression \(\sec^2 x\) using the identity: \(\sec^2 x = 1 + \tan^2 x\).
4. Compare this expression with the options in Column II to find the matching one.
5. Select the expression that matches \(1 + \tan^2 x\) as the correct completion of the identity.

Key Concepts

Pythagorean Identities

Pythagorean identities are fundamental trigonometric equations derived from the Pythagorean theorem. The most common is sin²x + cos²x = 1, which can be manipulated to express sec²x in terms of tan²x, forming the basis for many trigonometric simplifications.

Definition of Secant Function

The secant function, sec x, is defined as the reciprocal of the cosine function: sec x = 1/cos x. Understanding this reciprocal relationship helps in transforming and simplifying expressions involving sec²x.

Trigonometric Identity for sec²x

The identity sec²x = 1 + tan²x is a direct consequence of the Pythagorean identity and the definition of tangent. This identity is essential for completing expressions involving sec²x and is widely used in solving trigonometric equations.