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Chapter 6: Problem 78

In Exercises \(27-80,\) verify the given identities. $$b(\sec x-\tan x)(\sec x+\tan x)=b$$

Short Answer

Expert verified
Yes, the given identity \(b(\sec x-\tan x)(\sec x+\tan x)=b\) is indeed correct. The identity was verified using the difference of squares formula and the Pythagorean trigonometric identity.

Step by step solution

01

Simplify the Left Hand Side

The left hand side of the equation is \(b(\sec x-\tan x)(\sec x+\tan x)\), which can be simplified using the difference of squares formula \((a-b)(a+b) = a^2 - b^2\). Substituting \(a = \sec x\) and \(b = \tan x\), we get: \(b((\sec^2 x) - (\tan^2 x))\).
02

Apply the Pythagorean Identity

Next, apply the Pythagorean identity \(\sec^2 x - \tan^2 x = 1\), which simplifies the equation to be: \(b * 1 = b\).
03

Finalize the Simplification

Finally, further simplification yields \(b = b\), which verifies the given trigonometric identity.

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Key Concepts

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

Trigonometric Identities
Understanding trigonometric identities is fundamental to mastering many aspects of mathematics, particularly in the study of trigonometry and calculus. Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the equation are defined. These identities are crucial for simplifying expressions and solving trigonometric equations.

There are several basic classes of trigonometric identities, such as reciprocal identities, quotient identities, Pythagorean identities, even-odd identities, and sum and difference identities. For instance, reciprocal identities describe the relationship between the sine, cosine, and tangent functions and their reciprocals, the cosecant, secant, and cotangent functions, respectively. These identities are tools that can transform complex trigonometric expressions into simpler forms, enabling the solution of a wider array of problems.
Pythagorean Identity
The Pythagorean identity is one of the most well-known trigonometric identities, and it plays a central role in verifying other trigonometric identities like the one mentioned in the exercise. The fundamental Pythagorean identity is \( \sin^2 x + \cos^2 x = 1 \). This statement arises from the Pythagorean theorem applied to a right triangle with a radius (hypotenuse) of one (unit circle).

A derivative of this identity is used in the exercise solution: \( \sec^2 x - \tan^2 x = 1 \). The secant and tangent functions are the reciprocals of the cosine and sine functions, respectively. Because of this, the Pythagorean identity is also valid when expressed in terms of secant and tangent, providing a powerful tool for simplifying trigonometric expressions involving these functions. Recognizing when to apply the Pythagorean identity can make complex trigonometric problems much more manageable.
Difference of Squares
The difference of squares is a pattern used in algebra that is essential for factoring certain types of polynomials and also appears in trigonometric identities simplification, like in our example. It states that for any real numbers \(a\) and \(b\), the expression \(a^2 - b^2\) can be factored into \(a - b\) times \(a + b\). This is known as the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\).

In trigonometric terms, this is instrumental in decomposing expressions that may appear complex at first glance. By utilizing the format of the difference of squares, one can reveal underlying trigonometric identities that simplify the equation. As evidenced in the given exercise, recognizing the left-hand side as a difference of squares allowed for the straightforward application of the Pythagorean identity, leading to a simple verification of the trigonometric identity \(b = b\).

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Most popular questions from this chapter

An object is thrown upward at an acute angle \(\theta\) with respect to the horizontal. The object leaves the thrower's hand at a speed of 64 feet per second. Its horizontal coordinate \(x\) is given by \(x=(64 \cos \theta) t,\) where \(x\) is in feet and \(t\) is the time (in seconds) since the object was thrown. If \(x=64 \sqrt{3}\) when \(t=2 \sqrt{3},\) and the expression for \(x\) can be written as \(x=(64 \sin (\theta-c)) t\) for some \(c\) that \(0^{\circ} < \theta-c < 90^{\circ},\) find the values of \(\theta\) and \(c\) (in degrees).

Let \(\theta\) be the angle (in radians) that satisfies the conditions \(\cos \theta=-\frac{3}{5}\) and \(\pi<\theta<\frac{3 \pi}{2},\) and find the value of each. $$\cos \frac{\theta}{2}$$

In Exercises \(63-68,\) write in terms of a single trigonometric function of just \(x\). $$\tan \left(x+\frac{\pi}{2}\right)$$

The formula $$h(t)=125 \sin \left(2 \pi t-\frac{\pi}{2}\right)+125$$ represents the height above the ground at time \(t\), in minutes, of a person who is riding a ferris wheel. During the first turn, how much time does a passenger spend at or above a height of 200 feet?

Let \(\theta\) be the angle (in radians) that satisfies the conditions \(\cos \theta=-\frac{3}{5}\) and \(\pi<\theta<\frac{3 \pi}{2},\) and find the value of each. $$\sec \frac{\theta}{2}$$
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