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


II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan 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 \(\sec^2 x\) in terms of tangent using the identity: \(\sec^2 x = 1 + \tan^2 x\).
4. Compare the given expression \(\sec^2 x\) with the options in Column II to find the one that matches \(1 + \tan^2 x\).
5. Select the expression from Column II that completes the identity as \(1 + \tan^2 x\).

Key Concepts

Pythagorean Identity

The Pythagorean identities relate the squares of sine, cosine, and secant functions. One key identity is 1 + tan²x = sec²x, which expresses sec²x in terms of tan²x. This identity is fundamental for transforming and simplifying trigonometric expressions.

Definition of Secant Function

Secant (sec x) is the reciprocal of cosine, defined as sec x = 1/cos x. Understanding this reciprocal relationship helps in manipulating expressions involving sec²x and connecting them to other trigonometric functions like cosine and tangent.

Trigonometric Function Squares

Squaring trigonometric functions, such as sec²x or tan²x, is common in identities and equations. Recognizing how these squares relate through identities allows for simplification and solving of trigonometric problems effectively.