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Chapter 5: Problem 35

In Exercises 9-50, verify the identity \( \dfrac{\tan x \cot x}{\cos x} = \sec x \)

Short Answer

Expert verified
Yes the given trigonometric identity \( \dfrac{\tan x \cot x}{\cos x} = \sec x \) is correct.

Step by step solution

01

Substitute with Basic Trigonometric Functions

Replace tan(x), cot(x), cos(x) and sec(x) with their basic sine and cosine representation. So tan(x) = sin (x) / cos (x), cot(x) = cos (x) / sin (x), cos(x) = cos(x), sec(x) = 1/cos(x). The equation now looks as follows: \( \dfrac{\dfrac{\sin x}{\cos x} * \dfrac{\cos x}{\sin x}}{\cos x} = \dfrac{1}{\cos x} \)
02

Simplify the Expression

Notice how in the fraction that constitutes the left hand side of the equation, the factors sin(x) and cos(x) in the numerator cancel out with those in the denominator, leaving 1 in the numerator. \( \dfrac{1}{\cos x} = \dfrac{1}{\cos x} \)
03

Verify the Identity

At this point, the left-hand side of the equation is identical to the right-hand side of the equation, meaning that the proposed identity is verified and correct in all cases.

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Key Concepts

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

Trigonometric Functions
Understanding trigonometric functions is crucial in solving and verifying trigonometric identities. These functions鈥攕ine, cosine, tangent, cotangent, secant, and cosecant鈥攁re fundamentally related to the geometry of a circle.

Imagine a unit circle where any given angle's sine is the y-coordinate, and its cosine is the x-coordinate of the point where the terminal side of the angle intersects the circle. The tangent of an angle can be visualized as the slope of the line that extends from the origin to this point. These definitions hinge upon the unit circle, leading to various mathematical properties and relationships that are consistent across all angles.
Sine and Cosine
Sine and cosine are perhaps the most fundamental trigonometric functions. They provide a direct link between angles and the geometry of right triangles. For example, in a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse. In contrast, the cosine is the ratio of the adjacent side to the hypotenuse.

Understanding Ratios:
The sine of angle x, written as \( \text{sin}(x) \), is calculated as \( \frac{\text{opposite}}{\text{hypotenuse}} \) while the cosine of angle x, or \( \text{cos}(x) \), is \( \frac{\text{adjacent}}{\text{hypotenuse}} \).
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the included variables. They play a significant role in simplifying expressions and solving trigonometric equations. Some basic trigonometric identities include the Pythagorean identities, quotient identities, and reciprocal identities.

Utilizing Identities:
Key to verifying identities is knowing when and how to apply these relationships. For instance, in our exercise, by recognizing tangent as \( \text{sin}(x) / \text{cos}(x) \) and cotangent as its reciprocal, we can simplify complex fractions and verify the proposed identity.
Precalculus
Precalculus serves as the foundation for calculus, and a strong understanding of trigonometry is a cornerstone of precalculus. It's not just about solving identities; it's about conceptualizing how the trigonometric functions come together and apply to real-world scenarios.

Precalculus often involves manipulating and graphing functions, comprehending their properties, and solving equations algebraically. Students learn to visualize trigonometric functions not just as ratios or points on the unit circle, but as wave-like graphs representing periodic behavior, crucial for the advanced study of calculus.

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Most popular questions from this chapter

Consider the function given by \( f(x) = \sin^4 x + \cos^4 x \). (a) Use the power-reducing formulas to write the function in terms of cosine to the first power. (b) Determine another way of rewriting the function.Use a graphing utility to rule out incorrectly rewritten functions. (c) Add a trigonometric term to the function so that it becomes a perfect square trinomial. Rewrite the function as a perfect square trinomial minus the term that you added. Use a graphing utility to rule out incorrectly rewritten functions. (d) Rewrite the result of part (c) in terms of the sine of a double angle. Use a graphing utility to rule out incorrectly rewritten functions. (e) When you rewrite a trigonometric expression,the result may not be the same as a friends. Does this mean that one of you is wrong? Explain.

In Exercises 73-76, use the half-angle formulas to simplify the expression. \( \sqrt{\dfrac{1 + \cos 4x}{2}} \)

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( 7 \cos (-5 \beta) \sin 3 \beta \)

In Exercises 91-98, use the sum-to-product formulas to write the sum or difference as a product. \( \sin (\alpha + \beta) - \sin (\alpha - \beta) \)

In Exercises 73-76, use the half-angle formulas to simplify the expression. \( \sqrt{\dfrac{1 - \cos 8x}{1 + \cos 8x}} \)
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