Problem 18 Prove the identity. $$(\csc x-... [FREE SOLUTION]

Chapter 7: Problem 18

Prove the identity. $$(\csc x-1)(\csc x+1)=\cot ^{2} x$$

Short Answer

Expert verified

Based on the step-by-step solution provided, create a short-answer question: Question: Prove the trigonometric identity: $$(\csc x - 1)(\csc x + 1) = \cot^2 x$$ Answer: To prove the given trigonometric identity, we first convert the expression to sine and cosine functions. Then, we apply the difference of squares formula and find a common denominator. Next, we use the Pythagorean identity to simplify the expression. Finally, we convert back to the cotangent function to arrive at the desired identity: $$(\csc x - 1)(\csc x + 1) = \cot^2 x$$

Step by step solution

Convert to sine and cosine functions

Recall that $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$. Let's convert the given expression to sine and cosine functions: $$(\frac{1}{\sin x} - 1)(\frac{1}{\sin x} + 1)$$

Use the difference of squares formula

Now, let's apply the difference of squares formula to the expression: $a^2 - b^2 = (a - b)(a + b)$. In our case, let $a = \frac{1}{\sin x}$ and $b = 1$. Then, we get: $$a^2 - b^2 = \left(\frac{1}{\sin x}\right)^2 - 1^2 = \frac{1}{\sin^2 x} - 1$$

Find a common denominator

Since we're working with fractions, it's essential to find a common denominator. In this case, the common denominator is $\sin^2{x}$. We will rewrite our expression with the common denominator: $$\frac{1}{\sin^2 x} - 1 = \frac{1 - \sin^2 x}{\sin^2 x}$$

Apply the Pythagorean identity

One of the Pythagorean identities is $\sin^2 x + \cos^2 x = 1$. We can manipulate this equation to obtain an expression in terms of $\cos^2 x$: $$\cos^2{x} = 1 - \sin^2{x}$$ Now substitute this into our expression from Step 3: $$\frac{1 - \sin^2 x}{\sin^2 x} = \frac{\cos^2 x}{\sin^2 x}$$

Convert back to cotangent function

Recall that $\cot x = \frac{\cos x}{\sin x}$. Hence, $\cot^2 x = \frac{\cos^2 x}{\sin^2 x}$. Replace the expression in Step 4 with the cotangent function: $$\frac{\cos^2 x}{\sin^2 x} = \cot^2 x$$ So, we have proved the identity: $$(\csc x - 1)(\csc x + 1) = \cot^2 x$$

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Key Concepts

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

Sine and Cosine Functions

In the realm of trigonometry, the sine and cosine functions act as fundamental building blocks for describing the relationships between the angles and sides of a right triangle. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Similarly, the cosine is the ratio of the adjacent side's length to the hypotenuse's length.

To remember this, you can use the mnemonic SOHCAHTOA, which stands for Sine is Opposite over Hypotenuse; Cosine is Adjacent over Hypotenuse; Tangent is Opposite over Adjacent.

These functions also extend to the unit circle, where they represent the y-coordinate (sine) and x-coordinate (cosine) of a point on the circle related to a given angle. Hence, sine and cosine provide the foundation for solving trigonometric problems and proving identities, like the exercise provided.

Difference of Squares Formula

A powerful algebraic tool, the difference of squares formula, pivots on the fact that the product of two conjugate binomials (i.e., one with a plus and one with a minus, like $a+b$ and $a-b$) results in a simple form: $a^2 - b^2$. This is crucial in simplifying expressions and solving equations.

The difference of squares is applied not just in basic algebra but also intricately tied to higher-level mathematics, including trigonometric identities. By recognizing an expression that fits the form $a+b$ and $a-b$, we can swiftly move forward and simplify it without laborious multiplication. As seen in the expression:

$ \(a = \frac{1}{\text{sin} x}$ and $b = 1$\)

we acquire $a^2 - b^2$ after using the difference of squares, which considerably simplifies the problem at hand.

Pythagorean Identities

The Pythagorean identities in trigonometry emerge from the Pythagorean theorem, which links the sides of a right triangle. These identities are vital trigonometric equations incorporating sine and cosine functions.

The most prevalent Pythagorean identity you'll come across is $\text{sin}^2 x + \text{cos}^2 x = 1$. It is derived from the Pythagorean theorem's equation $a^2 + b^2 = c^2$, where $a$ and $b$ represent the sides of a right triangle and $c$ stands for the hypotenuse, considering that the triangle is inscribed in a unit circle.

All functions in trigonometry can somehow be related back to sine and cosine through these identities. Understanding and using the Pythagorean identities are crucial for proving more complex trigonometric identities, as observed in our exercise, where the identity

$\cos^2 x = 1 - \sin^2 x$

is used to prove the given trigonometric identity.

91影视

Prove the identity. $$(\csc x-1)(\csc x+1)=\cot ^{2} x$$

Short Answer

Step by step solution

Convert to sine and cosine functions

Use the difference of squares formula

Find a common denominator

Apply the Pythagorean identity

Convert back to cotangent function

Key Concepts

Sine and Cosine Functions

Difference of Squares Formula

\( \(a = \frac{1}{\text{sin} x}\) and \(b = 1\)\)

Pythagorean Identities

\(\cos^2 x = 1 - \sin^2 x\)

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