Question

Simplify, and then rewrite each expression as one of the three reciprocal trigonometric functions, csc \(x,\) sec \(x,\) or cot \(x\). a) \(\left(\frac{\cos x}{\tan x}\right)\left(\frac{\tan x}{\sin x}\right)\) b) \(\csc x \cot x \sec x \sin x\) c) \(\frac{\cos x}{1-\sin ^{2} x}\)

Short answer

a) \(\cot^2 x\), b) \(\csc x \), c) \(\sec x\).

Step-by-step solution

- Simplify expression a

Simplify \(\frac{\cos x}{\tan x}\frac{\tan x}{\sin x}\) by canceling \(\tan x\) in the numerator and denominator.

- Rewrite expression a

Since \(\tan x = \frac{\sin x}{\cos x}\), we have \(\frac{\cos x}{\frac{\sin x}{\cos x}}\frac{\frac{\sin x}{\cos x}}{\sin x} = \frac{\cos^2 x}{\sin^2 x} \). Therefore, \(\frac{\cos^2 x}{\sin^2 x} = \cot^2 x\).

- Simplify expression b

Rewrite \(\frac{1}{\sin x} \frac{\cos x}{\sin x} \frac{1}{\cos x} \sin x\) by canceling \(\sin x\) and \(\cos x\) where possible.

- Simplify further expression b

We are left with \(\frac{1}{\sin x} \cos x \sin x \frac{1}{\cos x} = 1\).

- Fix expression b in terms of trigonometric functions

Recall that \(\frac{1}{\sin x} = \csc x, \frac{1}{\cos x} = \sec x, \text{and } \frac{1}{\tan x} = \cot x\). Therefore, it simplifies to \(\csc x \text{ since } 1 = \csc x\) when rewritten in terms of \(\csc x\).

- Simplify expression c

Rewrite \(\frac{\cos x}{1 - \sin^2 x}\) using the Pythagorean trigonometric identity: \(1 - \sin^2 x = \cos^2 x\).

- Rewrite expression c

Since \(\frac{\cos x}{\cos^2 x} = \frac{1}{\cos x}\), expression c simplifies to \(\sec x \).

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are handy for simplifying expressions and solving trigonometric equations. Some common identities you should know include:

• Pythagorean Identity: \[\sin^2 x + \cos^2 x = 1\]
• Reciprocal Identities: \[ \csc x = \frac{1}{\sin x}, \sec x = \frac{1}{\cos x}, \cot x = \frac{1}{\tan x}\]

Recognizing and applying these identities will simplify complex expressions. For example, recognizing that \[1 - \sin^2 x = \cos^2 x\], allows us to simplify the given expression and convert it into a more understandable form.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves using identities to rewrite expressions in a simpler form. Here's how to do it step-by-step:
Remembering the reciprocal identities, rewrite \[\left(\frac{\cos x}{\tan x}\right)\left(\frac{\tan x}{\sin x}\right) = \frac{\cos x}{\frac{\sin x}{\cos x}} \cdot \frac{\frac{\sin x}{\cos x}}{\sin x}.\]
By canceling out common terms, we get \[\frac{\cos^2 x}{\sin^2 x} = \cot^2 x\].
For the expression \[\csc x \cot x \sec x \sin x\], it simplifies to 1 when we cancel out the terms, and hence rewrite it as \[\csc x\].
Lastly, \[\frac{\cos x}{1 - \sin^2 x}\] simplifies to \[\sec x\] by recognizing the Pythagorean identity.

Reciprocal Functions

Reciprocal functions in trigonometry are derived from the basic trigonometric functions. Here's a quick recap:

• \( \csc x = \frac{1}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \cot x = \frac{1}{\tan x} \)

By rewriting expressions using these reciprocal relationships, we can often simplify complex expressions. For example, if we want to rewrite \[\frac{1}{\sin x} \] in terms of a reciprocal function, we get \[\csc x\]. This technique simplifies the process of working through trigonometric problems.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry given by:

\[ \sin^2 x + \cos^2 x = 1\]
It helps us to relate \[\sin x\] and \[\cos x\]. For example, if an expression involves \[1 - \sin^2 x\], we can use the Pythagorean identity to rewrite it as \[\cos^2 x\]. This can significantly simplify complex expressions, like transforming \[\frac{\cos x}{1 - \sin^2 x}\] into \[\frac{\cos x}{\cos^2 x} = \sec x\]. Knowledge of this identity helps solve many trigonometric problems efficiently.