Problem 18 In Exercises I to $42,$ verify... [FREE SOLUTION]

Chapter 5: Problem 18

In Exercises I to $42,$ verify each identity. $$\frac{\sin x}{1+\cos x}=\csc x-\cot x$$

Short Answer

Expert verified

$\frac{\sin x}{1+\cos x}=\csc x-\cot x$ has been verified as a correct trigonometric identity. The steps involved rewriting $\csc x$ and $\cot x$ in terms of $\sin x$ and $\cos x$, then progressively simplifying the left-hand side of the equation to show it equals to the right-hand side.

Step by step solution

Rewrite Right Side of Equation

The first step is rewriting the right side of the equation using the definitions of cosecant and cotangent. Here we obtain: $\csc x-\cot x = \frac{1}{\sin x} - \frac{\cos x}{\sin x} = \frac{1-\cos x}{\sin x}$

Simplify Left Side of The Equation

Now, it is time to manipulate the left side of the equation to make it look like the right side. Since the denominator on the right is $\sin x$, on the left side, separate the fraction into two terms where $\sin x$ is in the denominator: $\frac{\sin x}{1+\cos x} = \frac{\sin x - \sin x\cos x }{(1+\cos x)} = \frac{\sin x(1- \cos x) }{(1+\cos x)}$. Get a common denominator (which is $1+ \cos x$) and this yields: $\frac{\sin x(1- \cos x) }{(1+\cos x)} = \frac{\sin x - \sin x\cos x }{(1+\cos x)}$

Destribution of $\sin x$ Over $1-\cos x$

Notify that $\sin x$ is both on the numerator on both sides. So distribute $\sin x$ over $1-\cos x$ and simplify. This yields: $\frac{\sin x - \sin x\cos x }{(1+\cos x)}$

Confirming Identity Eqality

Comparing both the left side and right side of the equation, we can see that they have the same expression, validating $\frac{\sin x}{1+\cos x}=\csc x-\cot x$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosecant Function

The cosecant function is one of the six standard trigonometric functions. It is the reciprocal of the sine function. So, if we have \ $ \sin x = a \ $, then the cosecant of $ x $ is given by: \\[ \csc x = \frac{1}{\sin x} \] \This means that the cosecant function gives the ratio of the hypotenuse to the opposite side in a right-angled triangle when you have an angle. Because it depends on the sine, it is undefined wherever the sine is zero.\

Easy to remember: "Cosecant" starts with "C," just like "Complimentary." It's more "complimentary" to talk about its sine.
Useful in finding angles and side lengths when one needs the opposite side or hypotenuse.

Understanding how cosecant relates to sine is important, especially when solving trigonometric equations or proofs where reciprocal identities come into play.

Cotangent Function

Similar to the cosecant function, cotangent is also an important trigonometric function. Cotangent is the reciprocal of the tangent function. With tangent known to be the ratio of sine to cosine, cotangent simply becomes the inverse: \\[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \]\This function gives the ratio of the adjacent side to the opposite side in a right-angled triangle when you have an angle. Like the cosecant function with sine, the cotangent function is undefined where the sine is zero.\

Recall: "Cotangent" also starts with a "C," indicating it's the reciprocal of "Tangent."
Helps to find the measure of angles and relations between sides within triangles.

Cotangent is particularly useful when simplifying and transforming trigonometric expressions, often in partnership with its inverse, tangent.

Simplifying Expressions

Simplifying trigonometric expressions is a critical skill in math. It often involves transforming one side of an equation to match the other using known identities or basic algebraic manipulation. This includes combining fractions, canceling terms, and using reciprocal and Pythagorean identities.\

Start by rewriting terms using basic definitions such as $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$.
Look for common terms: By finding a common denominator, you can often combine fractions into a simplified form.
Check each step: After altering expressions, always ensure each transformation is mathematically valid.

Simplifying helps in verifying identities, solving equations, and understanding underlying trigonometric properties efficiently.

Trigonometric Proofs

Trigonometric proofs require showing that one side of an equation is equivalent to the other, using known identities and logical steps. When dealing with trigonometric identities, it's crucial to start by manipulating one side, often the more complex one, until it resembles the other one. A typical strategy includes: \

Identifying which trigonometric identities apply. In our problem, reciprocals of sine and cosine are key.
Transforming expressions: For example, expressing $\csc x - \cot x$ in terms of $\sin x$ and $\cos x$.
Step-by-step logical sequence: Each transformation should follow from the previous logically and mathematically.

Trigonometric proofs help consolidate knowledge of the functions while providing a clear path to see connections between different concepts within trigonometry.

91影视

In Exercises I to \(42,\) verify each identity. $$\frac{\sin x}{1+\cos x}=\csc x-\cot x$$

Short Answer

Step by step solution

Rewrite Right Side of Equation

Simplify Left Side of The Equation

Destribution of \(\sin x\) Over \(1-\cos x\)

Confirming Identity Eqality

Key Concepts

Cosecant Function

Cotangent Function

Simplifying Expressions

Trigonometric Proofs

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