For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
sec x/csc 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 definitions of the secant and cosecant functions in terms of sine and cosine: \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\).
2. Rewrite the given expression \(\frac{\sec x}{\csc x}\) by substituting these definitions: \(\frac{\frac{1}{\cos x}}{\frac{1}{\sin x}}\).
3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: \(\frac{1}{\cos x} \times \frac{\sin x}{1} = \frac{\sin x}{\cos x}\).
4. Recognize that \(\frac{\sin x}{\cos x}\) is the definition of the tangent function, so the expression simplifies to \(\tan x\).
5. Therefore, the expression \(\frac{\sec x}{\csc x}\) is equivalent to \(\tan x\), which completes the identity.

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal functions relate basic trig functions to their inverses, such as sec x = 1/cos x and csc x = 1/sin x. Understanding these allows rewriting expressions like sec x/csc x in terms of sine and cosine for simplification.

Simplifying Trigonometric Expressions

Simplification involves rewriting expressions using fundamental identities and algebraic manipulation. For example, converting sec x/csc x into (1/cos x) ÷ (1/sin x) helps reduce the expression to a simpler form like sin x/cos x.

Trigonometric Identities

Identities such as tan x = sin x/cos x are essential for recognizing equivalent expressions. After simplification, sec x/csc x can be identified as tan x, completing the identity by matching expressions from both columns.