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Chapter 6: Problem 50

Prove that the equations are identities. $$\frac{1}{\sin \theta}-\sin \theta=\cot \theta \cos \theta$$

Short Answer

Expert verified
The equation is an identity because both sides simplify to \( \cot \theta \cos \theta \).

Step by step solution

01

Simplify Left Side of Equation

Start with the left side of the equation: \( \frac{1}{\sin \theta} - \sin \theta \).ewlineThis can be rewritten as: \( \frac{1 - \sin^2 \theta}{\sin \theta} \).ewlineWe know from the Pythagorean identity that \( 1 - \sin^2 \theta = \cos^2 \theta \).ewlineSubstitute in the identity: \( \frac{\cos^2 \theta}{\sin \theta} \).
02

Split Fraction

Take the expression \( \frac{\cos^2 \theta}{\sin \theta} \) from Step 1 and split it into: \( \frac{\cos \theta}{\sin \theta} \times \cos \theta \).ewlineThis can be simplified to: \( \cot \theta \cos \theta \).
03

Verify Both Sides Are Equal

Now, compare this result \( \cot \theta \cos \theta \) to the right side of the original equation.ewlineBoth expressions are \( \cot \theta \cos \theta \), confirming the equation is an identity.

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Key Concepts

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

Pythagorean Identity
The Pythagorean Identity is one of the key foundational concepts in trigonometry. It's derived directly from the Pythagorean Theorem, which is familiar to most students. The identity states that: \( \sin^2 \theta + \cos^2 \theta = 1 \). This is a crucial relationship because it holds true for all angles \( \theta \).
  • It allows us to express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \), as follows: \( \cos^2 \theta = 1 - \sin^2 \theta \).
  • This is particularly useful when simplifying trigonometric expressions or solving equations, as it can be used to replace one trigonometric function with another.
In our exercise, this identity is used to transform the expression \( 1 - \sin^2 \theta \) into \( \cos^2 \theta \). By substituting \( \cos^2 \theta \) into the fraction, we streamline the equation, simplifying our work when verifying the identity.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves rewriting them in a simpler or more convenient form. This process often makes solving equations easier.
  • The first step usually involves applying known identities, such as the Pythagorean Identity or other fundamental trigonometric identities.
  • In our example, we start with the expression \( \frac{1}{\sin \theta} - \sin \theta \) and use the identity to rewrite it as \( \frac{\cos^2 \theta}{\sin \theta} \).
  • By splitting the fraction \( \frac{\cos^2 \theta}{\sin \theta} \) into \( \frac{\cos \theta}{\sin \theta} \times \cos \theta \), we make the expression easier to manage.
This type of simplification can help us verify if two sides of an equation are indeed equal, allowing us to confirm whether an equation is an identity.
Cotangent Function
The cotangent function, denoted as \( \cot \theta \), is a less frequently used trigonometric function but still very important. It's defined as the reciprocal of the tangent function.
  • This function can be expressed as \( \cot \theta = \frac{1}{\tan \theta} \), which also equals \( \frac{\cos \theta}{\sin \theta} \).
  • In our solution, we see the use of \( \cot \theta \) when we simplify the fraction \( \frac{\cos \theta}{\sin \theta} \). This helps in rewriting our original trigonometric expression in terms of \( \cot \theta \).
  • The cotangent function is particularly helpful when simplifying expressions or equations that involve ratios of sine and cosine.
Understanding how to manipulate and utilize the cotangent function is vital, especially in proving identities or analyzing trigonometric functions more broadly.

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Most popular questions from this chapter

Write in terms of sine and cosine and simplify expression. $$(\sec A+\tan A)(\sec A-\tan A)$$

Four functions \(S, C, T,\) and \(D\) are defined as follows: \(\left.\begin{array}{l}S(\theta)=\sin \theta \\ C(\theta)=\cos \theta \\\ T(\theta)=\tan \theta \\ D(\theta)=2 \theta\end{array}\right\\} \quad 0^{\circ}<\theta<90^{\circ}\) In each case, use the values to decide if the statement is true or false. A calculator is not required. $$T\left(60^{\circ}\right)=2\left[T\left(30^{\circ}\right)\right]$$

Only one of the following two equations is an identity. Decide which equation this is, and give a proof to show that it is, indeed, an identity. For the other equation, give an example showing that it is not an identity. (For example, to show that the equation \(\sin \theta+\cos \theta=1\) is not an identity, let \(\theta=30^{\circ} .\) Then the equation becomes \(1 / 2+\sqrt{3} / 2=1\) which is false.) (a) \(\frac{\csc ^{2} \alpha-1}{\csc ^{2} \alpha}=\cos \alpha\) (b) \(\left(\sec ^{2} \alpha-1\right)\left(\csc ^{2} \alpha-1\right)=1\)

This exercise completes the discussion in the text concerning the use of parentheses in calculator work. (a) Use the unit circle definitions to briefly explain (in complete sentences) why \(\sin (\pi / 2)=1\) and \(\sin \pi=0\). (b) Set the calculator to the radian mode and enter the following sequence of keystrokes. $$\begin{array}{|c|c|c|c|c|} \hline \text { sin } & \pi & \div & 2 & \text { ENTER } \\ \hline \end{array}$$ Your calculator will show an output of \(0,\) which, as you know, is not the value of \(\sin (\pi / 2) .\) This is because, in the absence of parentheses, the calculator interprets the sequence of keystrokes \((\sin ) \quad \pi \quad(\div) \quad 2\) as follows: First compute sin \(\pi\), then divide the result by 2 That is, the calculator computes \(0 \div 2,\) which, of course, results in the 0 output. Conclusion: If you want the calculator to compute \(\sin (\pi / 2),\) you must use parentheses and enter the sequence of keystrokes

Use the given information to determine the values of the remaining five trigonometric functions. (The angles are assumed to be acute angles. ) $$\tan A=\frac{\sqrt{2}-1}{\sqrt{2}+1}$$
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