Textbook Question
Perform each indicated operation and simplify the result so that there are no quotients.
cos x/sec x + sin x/csc x
6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.RE.2
Textbook Question
For each expression in Column I, choose the expression from Column II that completes an identity.
2. 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 guidance1
Recall the definition of the cosecant function in terms of sine: \(\csc x = \frac{1}{\sin x}\).
Identify the expression in Column II that matches \(\frac{1}{\sin x}\) or is equivalent to it.
Verify that the chosen expression satisfies the identity by considering the domain where \(\sin x \neq 0\) to avoid division by zero.
Understand that this identity is fundamental because cosecant is the reciprocal of sine, which is a key concept in trigonometry.
Confirm that the expression from Column II correctly completes the identity \(\csc x = \frac{1}{\sin x}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Trigonometric Functions
The cosecant function (csc x) is the reciprocal of the sine function. This means csc x = 1/sin x, which is fundamental for rewriting or completing trigonometric identities involving csc x.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Recognizing and applying these identities helps simplify expressions and solve equations.
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Domain and Range of Trigonometric Functions
Understanding the domain and range of functions like sine and cosecant is important because csc x is undefined where sin x = 0. This knowledge ensures correct application of identities and avoids division by zero.
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