Problem 77 Prove the identity. $$\log _{1... [FREE SOLUTION]

Chapter 7: Problem 77

Prove the identity. $$\log _{10}(\csc x+\cot x)=-\log _{10}(\csc x-\cot x)$$

Short Answer

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Question: Prove the trigonometric/logarithmic identity $\log_{10}(\csc x+\cot x) = -\log_{10}(\csc x-\cot x)$. Answer: To prove this identity, we first replace $\csc x$ and $\cot x$ with their equivalent reciprocal identities as $\frac{1}{\sin x}$ and $\frac{\cos x}{\sin x}$, respectively. Then, we use logarithmic properties to express the difference of logarithms as a ratio. By simplifying the equation further, we show that the simplified expression inside the logarithm results in 1. Since the logarithm of 1 in any base is always 0, the identity is proven to be true: $\log _{10}(\csc x+\cot x)=-\log _{10}(\csc x-\cot x)$.

Step by step solution

Replace trigonometric identities

Write $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$ so that our equation becomes: $$\log_{10}\left(\frac{1+\cos x}{\sin x}\right)=-\log_{10}\left(\frac{1-\cos x}{\sin x}\right)$$

Use logarithmic properties

Using logarithmic base and exponent properties, we will get the equation in a more manageable form. First, we will express the difference of logarithms as a ratio: $$\log_{10}\frac{\left(\frac{1+\cos x}{\sin x}\right)}{\left(\frac{1-\cos x}{\sin x}\right)}=0$$

Simplify the equation

Now, we want to simplify the expression inside the logarithm: $$\frac{\left(\frac{1+\cos x}{\sin x}\right)}{\left(\frac{1-\cos x}{\sin x}\right)}=\frac{(1+\cos x)(\sin x)}{(1-\cos x)(\sin x)}$$ Dividing both the numerator and the denominator by $(\sin x)$ gives: $$\frac{1+\cos x}{1-\cos x}$$

Prove that the expression results in 1

To prove that the expression is equal to 0, we need to show that the simplified expression is equal to 1. Then, the logarithm of 1 in any base is always 0: $$\frac{1+\cos x}{1-\cos x}$$ Now, multiply the numerator and denominator by the conjugate of the denominator $(1+\cos x)$: $$\frac{(1+\cos x)^2}{(1-\cos x)(1+\cos x)}$$ Thus, we obtain: $$\frac{1+2\cos x+\cos^2 x}{1-\cos^2 x}$$ Using the Pythagorean identity $\cos^2 x+\sin^2 x = 1$ and solving for $\cos^2 x$ as $1-\sin^2 x$, we can substitute this into the above expression: $$\frac{1+2\cos x+(1-\sin^2 x)}{1-(1-\sin^2 x)}$$ Simplifying the above expression gives: $$\frac{2+2\cos x-\sin^2 x}{\sin^2 x}$$ Recognizing the expression $\sin^2 x$ in the numerator, we can rewrite the expression using Pythagorean identity: $$\frac{2+2\cos x-(1-\cos^2 x)}{\sin^2 x}$$ Simplifying further, we find: $$\frac{1+2\cos x+\cos^2 x}{\sin^2 x}$$ Now, substitute $\cos^2 x$ with $1-\sin^2 x$ again and then simplify: $$\frac{1+2\cos x+(1-\sin^2 x)}{\sin^2 x}$$ Cancelling out the terms, we find that the expression simplifies to 1: $$\frac{\sin^2 x}{\sin^2 x}=1$$

Conclude the proof

Since the simplified expression is equal to 1, the logarithm of the expression in base 10 is equal to 0: $$\log_{10}\left(\frac{1+\cos x}{1-\cos x}\right)=0$$ And our identity is proven: $$\log _{10}(\csc x+\cot x)=-\log _{10}(\csc x-\cot x)$$

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which both sides of the identity are defined. They are crucial in simplifying expressions and solving equations that involve trigonometric functions. A common example of a trigonometric identity is the Pythagorean identity, which states that $ \cos^2 x + \sin^2 x = 1 $.

In the context of the given exercise, we utilize the definition of cosecant $ \csc x = \frac{1}{\sin x} $ and cotangent $ \cot x = \frac{\cos x}{\sin x} $ as part of the process of proving a trigonometric identity. These initial substitutions allow for the simplification of the expression in subsequent steps, showcasing how an understanding of these fundamental identities is vital for manipulating and solving trigonometric equations.

Logarithmic Properties

Logarithmic properties are rules that describe how to manipulate logarithmic expressions. Understanding these rules is essential for solving logarithmic equations and simplifying logarithmic expressions. The properties include the product, quotient, and power rules, among others. For example, the quotient rule for logarithms states that $ \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) $, which was employed in Step 2 of our solution to combine the logs with negative sign into a single logarithm expressing a ratio.

Using logarithmic properties effectively can convert complex logarithmic expressions into more straightforward forms, thereby making difficult problems more manageable. In our proof, applying the quotient rule helps to set the stage for further simplification, which eventually leads to showing that the logarithm is of 1, which is equivalent to zero.

Proof Techniques in Precalculus

Proof techniques in precalculus involve logical reasoning to establish the truth of mathematical statements. Direct proofs, proof by contradiction, and proof by induction are common approaches. The process typically involves a series of logical steps where each step is justified by a mathematical principle or a previously established result.

In our example, the step-by-step approach effectively demonstrates a direct proof. We start with the identity we are trying to prove and manipulate it stepwise using known identities, definitions, and properties until we reach a conclusion that is obviously true (in this case, that $ \log_{10}(1) = 0 $). This process of direct proof is a foundational method in mathematics, allowing us to reason from known truths to establish new ones.

Pythagorean Identity

The Pythagorean identity is one of the most well-known and widely used trigonometric identities, which states that $ \sin^2 x + \cos^2 x = 1 $. This identity is derived from the Pythagorean theorem relating the sides of a right triangle and forms the basis for many other identities and results in trigonometry.

In the given exercise, we apply the Pythagorean identity to transform expressions involving $ \cos^2 x $ and $ \sin^2 x $, which simplifies the complex algebraic fraction within the logarithm. Recognizing when and how to apply the Pythagorean identity is a critical skill in solving trigonometric equations. By substituting $ \cos^2 x $ with $ 1 - \sin^2 x $ and vice versa, we are able to cancel terms and eventually simplify the expression to 1, leading to the conclusion of the proof. Understanding this identity allows students to navigate trigonometric problems with more ease and confidence.

91影视

Prove the identity. $$\log _{10}(\csc x+\cot x)=-\log _{10}(\csc x-\cot x)$$

Short Answer

Step by step solution

Replace trigonometric identities

Use logarithmic properties

Simplify the equation

Prove that the expression results in 1

Conclude the proof

Key Concepts

Trigonometric Identities

Logarithmic Properties

Proof Techniques in Precalculus

Pythagorean Identity

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