Problem 69
Question
Rewrite each expression in terms of the given function or functions. \(\frac{\cos x}{1+\sin x}+\tan x ; \cos x\)
Step-by-Step Solution
Verified Answer
The expression \(\frac{\cos x}{1+\sin x}+\tan x\) can be rewritten in terms of \( \cos x\) as \(\frac{\cos^2(x)}{2\cos(x)\sqrt{1 - \cos^2(x)}} + \frac{\sqrt{1 - \cos^2(x)}}{\cos x}\).
1Step 1: Recall Trigonometric Identities
Recall and apply the trigonometric identities: \(1 = \sin^2(x) + \cos^2(x)\), \(\tan(x) = \sin(x) / \cos(x)\). Accordingly, \(\sin x\) can be replaced by \(\sqrt{1 - \cos^2(x)}\), which results in the equation: \(\frac{\cos x}{1+\sqrt{1 - \cos^2(x)}} + \frac{\sqrt{1 - \cos^2(x)}}{\cos x}\).
2Step 2: Simplify the Resulting Expressions
Simplify the equation by multiplying the fractions with the suitable expressions to get rid of the roots in the denominator. This leads to: \(\frac{\cos^2(x)}{\cos^2(x) + 2\cos(x)\sqrt{1 - \cos^2(x)} + 1 - \cos^2(x)} + \frac{1 - \cos^2(x)}{\cos x}\).
3Step 3: Further Simplification
Simplify the equation further by rewriting the denominator and cancelling out terms. The final expression is: \(\frac{\cos^2(x)}{2\cos(x)\sqrt{1 - \cos^2(x)}} + \frac{\sqrt{1 - \cos^2(x)}}{\cos x}\).
Key Concepts
Simplifying ExpressionsCosine FunctionTangent Function
Simplifying Expressions
Trigonometric expressions can often appear complex, but simplifying them involves breaking them down using known identities. When you see an expression like \(\frac{\cos x}{1+\sin x} + \tan x\), you should first consider how to express all parts in terms of the simplest functions possible.
A critical step in this process is recognizing identities that allow substitution or factoring out common elements. In our case, the Pythagorean identity \(1 = \sin^2(x) + \cos^2(x)\) is helpful. By expressing \(\sin x\) in terms of \(\cos x\), specifically \(\sin x = \sqrt{1-\cos^2(x)}\), we can rewrite parts of the expression.
The goal is to clear up the square roots and simplify fractions where possible. Multiplying by conjugates or matching denominators can aid in this simplification. Always aim to reduce complex fractions to simpler terms, helping reveal simpler relationships or functions.
A critical step in this process is recognizing identities that allow substitution or factoring out common elements. In our case, the Pythagorean identity \(1 = \sin^2(x) + \cos^2(x)\) is helpful. By expressing \(\sin x\) in terms of \(\cos x\), specifically \(\sin x = \sqrt{1-\cos^2(x)}\), we can rewrite parts of the expression.
The goal is to clear up the square roots and simplify fractions where possible. Multiplying by conjugates or matching denominators can aid in this simplification. Always aim to reduce complex fractions to simpler terms, helping reveal simpler relationships or functions.
Cosine Function
The cosine function is one of the primary trigonometric functions and is often used to express other trigonometric functions. In the given problem, writing everything in terms of \(\cos x\) allows for a unified approach to simplifying the expression.
When you express \(\sin x\) in terms of \(\cos x\), as \(\sin x = \sqrt{1-\cos^2(x)}\), it allows us to address the square roots that complicate our expressions. This transformation is particularly useful because it restricts the function to angles where cosine is easier to manipulate or calculate.
Moreover, using cosine can help to evaluate the function’s behavior over its range, from -1 to 1. Understanding how to manipulate identities and expressions to primarily involve \(\cos x\) helps streamline many problems in trigonometric simplifications.
When you express \(\sin x\) in terms of \(\cos x\), as \(\sin x = \sqrt{1-\cos^2(x)}\), it allows us to address the square roots that complicate our expressions. This transformation is particularly useful because it restricts the function to angles where cosine is easier to manipulate or calculate.
Moreover, using cosine can help to evaluate the function’s behavior over its range, from -1 to 1. Understanding how to manipulate identities and expressions to primarily involve \(\cos x\) helps streamline many problems in trigonometric simplifications.
Tangent Function
The tangent function, defined as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), provides a bridge between sine and cosine. It can help simplify expressions by transforming division into multiplication when combined strategically with other trigonometric identities.
In our problem, \(\tan x\) is rewritten in terms of sine and cosine to integrate it coherently with our expression in terms of \(\cos x\). By simplifying \(\tan x\) directly using this identity, we reduce the complexity of the expression, allowing us to manipulate and simplify the terms more easily.
It's essential to recognize when \(\tan x\) can offer simplification because it can also be expressed in terms of \(\sin\) and \(\cos\) to provide additional ways to either simplify or uncover key features of the expression.
In our problem, \(\tan x\) is rewritten in terms of sine and cosine to integrate it coherently with our expression in terms of \(\cos x\). By simplifying \(\tan x\) directly using this identity, we reduce the complexity of the expression, allowing us to manipulate and simplify the terms more easily.
It's essential to recognize when \(\tan x\) can offer simplification because it can also be expressed in terms of \(\sin\) and \(\cos\) to provide additional ways to either simplify or uncover key features of the expression.
Other exercises in this chapter
Problem 69
Rewrite each expression as a simplified expression containing one term. $$ \cos (\alpha+\beta) \cos \beta+\sin (\alpha+\beta) \sin \beta $$
View solution Problem 69
Use an identity to solve each equation on the interval \([0,2 \pi)\) $$ \sin 2 x=\cos x $$
View solution Problem 70
Rewrite each expression as a simplified expression containing one term. $$ \sin (\alpha-\beta) \cos \beta+\cos (\alpha-\beta) \sin \beta $$
View solution Problem 70
Use an identity to solve each equation on the interval \([0,2 \pi)\) $$ \sin 2 x=\sin x $$
View solution