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Chapter 7: Problem 411

For the following exercises, prove the identity. $$ \cot x \cos (2 x)=-\sin (2 x)+\cot x $$

Short Answer

Expert verified
The identity is proven true as both sides simplify to \( \frac{\cos(x)\cos(2x)}{\sin(x)} \).

Step by step solution

01

Write down the Trigonometric Identities

Recall the double angle formulas for cosine and sine. We have \( \cos(2x) = \cos^2(x) - \sin^2(x) \) and \( \sin(2x) = 2\sin(x)\cos(x) \). Additionally, remember that \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
02

Substitute Identities into LHS

Substitute the identities for \( \cot(x) \) and \( \cos(2x) \) into the left-hand side of the equation. This gives us:\[ \cot(x)\cos(2x) = \frac{\cos(x)}{\sin(x)} (\cos^2(x) - \sin^2(x)) \].
03

Simplify LHS Expression

Distribute \( \frac{\cos(x)}{\sin(x)} \) over \( \cos^2(x) - \sin^2(x) \):\[ \frac{\cos^3(x)}{\sin(x)} - \frac{\cos(x)\sin^2(x)}{\sin(x)} \].Simplify the terms:\[ \frac{\cos^3(x)}{\sin(x)} - \cos(x)\sin(x) \].
04

Express LHS in Terms of Sine and Cosine

Now write both terms under a common denominator:\[ \frac{\cos^3(x) - \cos(x)\sin^2(x)}{\sin(x)} \].The numerator can be rewritten using \( \cos^3(x) - \cos(x)\sin^2(x) = \cos(x)(\cos^2(x) - \sin^2(x)) = \cos(x)\cos(2x) \), confirming our steps so far are consistent.
05

Write down the Right-Hand Side (RHS)

The right-hand side of the identity given is \(-\sin(2x) + \cot(x)\). Substitute the identities:\[ -2\sin(x)\cos(x) + \frac{\cos(x)}{\sin(x)} \].
06

Simplify and Compare RHS

Simplify \(-2\sin(x)\cos(x) + \frac{\cos(x)}{\sin(x)} \) by expressing with common denominators:\[ \frac{-2\sin^2(x)\cos(x) + \cos(x)}{\sin(x)} = \frac{-\cos(x)(2\sin^2(x) - 1)}{\sin(x)} \].Recognize \(-2\sin^2(x) + 1 = -\cos(2x) \), so rewrite:\[ \frac{\cos(x)\cos(2x)}{\sin(x)} \], which matches the left-hand side.

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Key Concepts

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

Double Angle Formulas
One of the essential sets of identities in trigonometry is the Double Angle Formulas. These formulas express trigonometric functions of \(2x\) in terms of \(x\), making them invaluable for various mathematical proofs and simplifications.

For the functions cosine and sine, the double angle formulas are:
\[ \cos(2x) = \cos^2(x) - \sin^2(x) \]
and
\[ \sin(2x) = 2\sin(x)\cos(x) \].

Each formula plays a vital role in solving trigonometric equations. For example, these transformations turn complicated expressions into more manageable ones, simplifying the process of proving trigonometric identities.
  • The formula for cosine allows us to see the relationship between the squares of sine and cosine.
  • The formula for sine shows a connection involving multiplication of the sine and cosine functions.

Mastering these transformations gives a powerful toolset to unravel the mysteries of complex trigonometric expressions. Remember, consistent practice with these identities will solidify your understanding.
Sine and Cosine Identities
Sine and cosine identities form the backbone of trigonometry. These basic formulas are key to solving more complex trigonometric equations and proofs.

First, recall the two fundamental identities:
  • \( \sin^2(x) + \cos^2(x) = 1 \) (Pythagorean identity)
  • \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

These identities set the stage for further expansion and manipulation of trigonometric expressions.

Applying these identities enables simplification of problems, like converting from one function to another, or finding the values of unknown angles and lengths in triangles. For instance, substituting \( \cot(x) \) with its identity gives a clearer path to simplifying complex trigonometric problems.
With practice, using sine and cosine identities becomes second nature, making complex trigonometry more approachable.
Cotangent Identity
The cotangent identity connects the cotangent function, often used to simplify trigonometric equations. It is defined as:
\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \].

This identity provides a way to express cotangent in terms of sine and cosine, which are more commonly used functions.
  • By expressing \( \cot(x) \) as a ratio of cosine to sine, it allows transformation and simplification of expressions.
  • Using the cotangent identity can effectively simplify proofs, especially when combined with other trigonometric identities.

Consider its application in proving other identities; for instance, in the problem provided, substituting \( \cot(x) \) with \( \frac{\cos(x)}{\sin(x)} \) plays a key role in reaching the proof efficiently.
Understanding the cotangent identity is crucial for those tricky problems where direct methods might be cumbersome.

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Most popular questions from this chapter

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator. Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 8 times per second, was initially pulled down 32 cm from equilibrium, and the amplitude decreases by 50% each second. The second spring, oscillating 18 times per second, was initially pulled down 15 cm from equilibrium and after 4 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider 鈥渞est鈥� as an amplitude less than 0.1 cm.

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For the following exercises, construct an equation that models the described behavior. A deer population oscillates 19 above and below average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 each year. Find a function that models the population, \(P,\) in terms of months since January, \(t\) .

For the following exercises, use basic identities to simplify the expression. $$ \sin ^{3} x+\cos ^{2} x \sin x $$

For the following exercises, find a function of the form \(y=a b^{x}+c \sin \left(\frac{\pi}{2} x\right)\) that fits the given data. $$ \begin{array}{|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} \\ \hline y & {6} & {29} & {96} & {379} \\ \hline\end{array} $$
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