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Chapter 6: Problem 51

Verify each of the trigonometric identities. $$\frac{1}{\csc ^{2} x}+\frac{1}{\sec ^{2} x}=1$$

Short Answer

Expert verified
The identity is verified using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \).

Step by step solution

01

Convert to Sine and Cosine

To simplify the given identity, we need to express the cosecant and secant functions in terms of sine and cosine. We know these relations: \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \). Therefore, \( \csc^2 x = \frac{1}{\sin^2 x} \) and \( \sec^2 x = \frac{1}{\cos^2 x} \). Rewrite the identity as: \[ \frac{1}{\frac{1}{\sin^2 x}} + \frac{1}{\frac{1}{\cos^2 x}} = 1. \] This simplifies to: \[ \sin^2 x + \cos^2 x = 1. \]
02

Use the Pythagorean Identity

Recall the fundamental Pythagorean identity in trigonometry: \( \sin^2 x + \cos^2 x = 1 \). Observing the right-hand side of our equation, we see it matches this identity exactly. This confirms that the original expression simplifies to the true identity, thus verifying it.

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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 an essential part of trigonometry, especially when dealing with reciprocal identities. It is the reciprocal of the sine function. In simpler terms, if you know the sine of an angle, you can find its cosecant by taking the reciprocal:\[ \csc x = \frac{1}{\sin x} \]This means that if \( \sin x = \frac{1}{2} \), then the cosecant would be \( \csc x = 2 \).
  • Inverse Relationship: The sine function and its reciprocal, the cosecant, are closely linked. The cosecant will not be defined for angles where \( \sin x = 0 \), as dividing by zero is not possible.
  • Key Angles: For angles like 0, 90, 180 degrees (multiples of 90), the cosecant function becomes undefined or extreme, due to the sine values approaching zero.
Understanding the cosecant function helps in transforming and verifying various trigonometric identities, making complex expressions easier to handle.
Secant Function
The secant function is very similar to the cosecant, but it is related to the cosine function. It is the reciprocal of the cosine, providing crucial insights into the behavior of trigonometric functions:\[ \sec x = \frac{1}{\cos x} \]For instance, if \( \cos x = \frac{1}{3} \), then \( \sec x = 3 \).
  • Reciprocal Nature: Just like the relationship between sine and cosecant, cosine and secant are each other's reciprocals. The secant function is undefined when \( \cos x = 0 \), which occurs at odd multiples of 90 degrees.
  • Graph Characteristics: The secant graph features vertical asymptotes where cosine equals zero, diving towards infinity at these points.
Being familiar with the secant function helps in simplifying trigonometric expressions and understanding their geometrical interpretations.
Pythagorean Identity
The Pythagorean identity is a foundational formula in trigonometry. It establishes a relationship between the squares of sine and cosine functions:\[ \sin^2 x + \cos^2 x = 1 \]This identity not only helps in verifying complex equations but is also critical in solving numerous trigonometric problems.
  • Origin: The identity derives from the Pythagorean theorem applied to a unit circle, where the hypotenuse is the radius of length 1.
  • Application: It is often used to simplify trigonometric expressions and convert between different forms, as seen in the original exercise where \( \sin^2 x + \cos^2 x = 1 \) helped verify the identity involving cosecant and secant functions.
Understanding this identity allows for mastering trigonometric proofs and computations efficiently.

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Most popular questions from this chapter

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$4 \cos ^{2} x-4 \sin x=5$$

Graphing calculators can be used to find approximate solutions to trigonometric equations. For the equation \(f(x)=g(x),\) let \(Y_{1}=f(x)\) and \(Y_{2}=g(x) .\) The \(x\) -values that correspond to points of intersections represent solutions. With a graphing utility, solve the equation \(\cos \theta=\csc \theta\) on \(0 \leq \theta \leq \pi\).

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$\csc x+\cot x=\sqrt{3}$$

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$-\frac{1}{4} \csc \left(\frac{1}{2} x\right)=\cos \left(\frac{1}{2} x\right)$$

Solve each trigonometric equation on \(0^{\circ} \leq \theta<360^{\circ} .\) Express solutions in degrees and round to two decimal places. $$\sec ^{2} x=\tan x+1$$
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