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Chapter 6: Problem 13

Verify the identity. $$\csc ^{4} t-\cot ^{4} t=\csc ^{2} t+\cot ^{2} t$$

Short Answer

Expert verified
The identity is verified: both sides equal \( \csc^2 t + \cot^2 t \).

Step by step solution

01

Rewrite the Left Side

Start by expressing the left side of the identity, \( (\csc^4 t - \cot^4 t) \), as a difference of squares. This can be written as:\[ (\csc^2 t)^2 - (\cot^2 t)^2 = (\csc^2 t - \cot^2 t)(\csc^2 t + \cot^2 t) \]
02

Recall the Pythagorean Identity

Recall the trigonometric identity: \( \csc^2 t = 1 + \cot^2 t \). Substituting this into \( \csc^2 t - \cot^2 t \), we have:\[ \csc^2 t - \cot^2 t = (1 + \cot^2 t) - \cot^2 t = 1 \]
03

Simplify Left Side Further

Using the results from Step 2 in the expression for the left side:\[ (\csc^2 t - \cot^2 t)(\csc^2 t + \cot^2 t) = (1)(\csc^2 t + \cot^2 t) = \csc^2 t + \cot^2 t \]
04

Compare Both Sides

Now, compare the simplified left side \( \csc^2 t + \cot^2 t \) with the right side of the identity, which is also \( \csc^2 t + \cot^2 t \). They are equal, verifying the identity.

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Key Concepts

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

Difference of Squares
The difference of squares is a powerful algebraic concept. It refers to a specific type of algebraic expression where you have the subtraction of two squared terms. For example, the expression \( a^2 - b^2 \) can be factored into \((a - b)(a + b)\). This factorization is beneficial when simplifying complex problems. In our case with the identity \( \csc^4 t - \cot^4 t \), we can set \( a = \csc^2 t \) and \( b = \cot^2 t \). By applying the difference of squares formula, we rewrite it as:
  • \( (\csc^2 t - \cot^2 t)(\csc^2 t + \cot^2 t) \)
This tactic makes it much easier to work with trigonometric expressions, allowing us to apply other identities effectively.
Pythagorean Identity
The Pythagorean identity is a fundamental trigonometric identity that connects the functions sine, cosine, cosecant, and cotangent. In particular, one of the identities is \( \csc^2 t = 1 + \cot^2 t \), derived from the basic identity of \( \sin^2 t + \cos^2 t = 1 \). This connection is vital when simplifying trigonometric expressions.In our context, we use \( \csc^2 t = 1 + \cot^2 t \) to transform and simplify the expression \( \csc^2 t - \cot^2 t \). By substituting for \( \csc^2 t \), we get:
  • \( 1 + \cot^2 t - \cot^2 t \)
  • This simplifies to \( 1 \)
This step is crucial as it reveals a simple multiplicative factor, helping us verify the identity with ease.
csc and cot Functions
Understanding the functions \( \csc t \) (cosecant) and \( \cot t \) (cotangent) is essential for solving trigonometric identities. These functions are reciprocals of sine and tangent, respectively.
  • The cosecant function is defined as \( \csc t = \frac{1}{\sin t} \)
  • The cotangent function is defined as \( \cot t = \frac{\cos t}{\sin t} \)
By knowing these definitions, you can express and manipulate trigonometric identities more straightforwardly.For the identity \( \csc^4 t - \cot^4 t = \csc^2 t + \cot^2 t \), recognizing and applying these functions correctly simplifies the involve expressions. Each term can easily relate back to sine and cosine, allowing easier application of identities such as the Pythagorean identity.

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Most popular questions from this chapter

For certain applications in electrical engineering, the sum of several voltage signals or radio waves of the same frequency is expressed in the compact form \(y=A \cos (B t-C) .\) Express the given signal in this form. $$y=50 \sin 60 \pi t+40 \cos 60 \pi t$$

Find the solutions of the equation that are in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$\cos t-\sin 2 t=0$$

Graphically solve the trigonometric equation on the indicated interval to two decimal places. $$\sec (2 x+1)=\cos \frac{1}{2} x+1 ; \quad[-\pi / 2, \pi / 2]$$

Verify the identity. $$\tan \frac{\theta}{2}=\csc \theta-\cot \theta$$

A graph of \(y=\cos 2 x+2 \cos x\) for \(0 \leq x \leq 2 \pi\) is shown in the figure. (a) Approximate the \(x\) -intercepts to two decimal places. (b) The \(x\) -coordinates of the turning points \(P, Q,\) and \(R\) on the graph are solutions of \(\sin 2 x+\sin x=0 .\) Find the coordinates of these points. (GRAPH CANNOT COPY)
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