Problem 91
Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning. The most efficient way that I can simplify \(\frac{(\sec x+1)(\sec x-1)}{\sin ^{2} x}\) is to immediately rewrite the expression in terms of cosines and sines.
Step-by-Step Solution
Verified Answer
The statement makes sense. Rewriting the equation in terms of cosine and sine and substituting identities simplifies the expression effectively to 1.
1Step 1: Evaluate the Statement
Original statement: The most efficient way that I can simplify \( \frac{(\sec x+1)(\sec x-1)}{\sin ^{2} x} \) is to immediately rewrite the expression in terms of cosines and sines.
2Step 2: Substitute Trigonometric Identities
As secant (sec) is the reciprocal of cosine, i.e., \( \sec x = \frac{1}{\cos x} \), and since sine squared plus cosine squared equals one (Pythagorean identity), i.e., \( \sin ^{2} x + \cos ^{2} x = 1 \) or equivalently \( \sin ^{2} x = 1 - \cos ^{2} x \), substitute these identities to rewrite the given expression.
3Step 3: Apply Algebraic Simplification
The expression becomes \( \frac{(\frac{1}{\cos x}+1)* (\frac{1}{\cos x}-1)}{(1-\cos ^{2} x)} \). After expanding the numerator and simplifying, the equation simplifies to 1.
Key Concepts
Algebraic SimplificationReciprocal IdentitiesPythagorean IdentityTrigonometric Functions
Algebraic Simplification
Algebraic simplification is the process of reducing mathematical expressions to a simpler form. It's all about breaking things down step by step. This makes it easier to solve or understand mathematical equations.
When we look at the expression \[ \frac{(\sec x+1)(\sec x-1)}{\sin^{2} x} \]we can simplify it by applying transformations and reducing terms. The goal is to use rules and identities to make the expression more manageable. Pay attention to things like:
Check for opportunities to factorize, expand or condense expressions, as seen with the original expression, where expanding and simplifying the numerator helps to convert it into something much simpler.
When we look at the expression \[ \frac{(\sec x+1)(\sec x-1)}{\sin^{2} x} \]we can simplify it by applying transformations and reducing terms. The goal is to use rules and identities to make the expression more manageable. Pay attention to things like:
- Removing like terms
- Using basic arithmetic operations
- Applying mathematical identities
Check for opportunities to factorize, expand or condense expressions, as seen with the original expression, where expanding and simplifying the numerator helps to convert it into something much simpler.
Reciprocal Identities
Reciprocal identities are special trigonometric identities that express functions as reciprocals of others. For instance, the secant (\( \sec x \)) function is the reciprocal of the cosine function:
\[ \sec x = \frac{1}{\cos x} \]
By understanding and using reciprocal identities, you can transform trigonometric expressions into different forms. This often makes them easier to handle, especially when solving equations or simplifying complex expressions.
In our exercise, we transform secant to cosine using this identity:
\[ (\sec x + 1)(\sec x - 1) \rightarrow \left(\frac{1}{\cos x} + 1\right) \left(\frac{1}{\cos x} - 1\right) \]
These transformations reveal simpler forms like the difference of squares or other manageable mathematical structures. Always remember, using reciprocal identities can be a powerful tool in trigonometry.
\[ \sec x = \frac{1}{\cos x} \]
By understanding and using reciprocal identities, you can transform trigonometric expressions into different forms. This often makes them easier to handle, especially when solving equations or simplifying complex expressions.
In our exercise, we transform secant to cosine using this identity:
\[ (\sec x + 1)(\sec x - 1) \rightarrow \left(\frac{1}{\cos x} + 1\right) \left(\frac{1}{\cos x} - 1\right) \]
These transformations reveal simpler forms like the difference of squares or other manageable mathematical structures. Always remember, using reciprocal identities can be a powerful tool in trigonometry.
Pythagorean Identity
The Pythagorean identity is one of the most important trigonometric identities. It relates the square of sine and cosine functions:
\[ \sin^{2} x + \cos^{2} x = 1 \]
This identity can be rearranged to solve for either sine squared or cosine squared. In our expression simplification, it helps to substitute \( \sin^{2} x \) with \( 1 - \cos^{2} x \). This step allows us to express everything in terms of cosine, which often makes further simplification easier.
Using the identity, the denominator becomes:
\[ \sin^{2} x = 1 - \cos^{2} x \]
Understanding and using the Pythagorean identity is crucial in mathematics because it connects different trigonometric functions, showing relationships that simplify many problems. It’s a go-to rule for decomposing trig functions into more workable forms.
\[ \sin^{2} x + \cos^{2} x = 1 \]
This identity can be rearranged to solve for either sine squared or cosine squared. In our expression simplification, it helps to substitute \( \sin^{2} x \) with \( 1 - \cos^{2} x \). This step allows us to express everything in terms of cosine, which often makes further simplification easier.
Using the identity, the denominator becomes:
\[ \sin^{2} x = 1 - \cos^{2} x \]
Understanding and using the Pythagorean identity is crucial in mathematics because it connects different trigonometric functions, showing relationships that simplify many problems. It’s a go-to rule for decomposing trig functions into more workable forms.
Trigonometric Functions
Trigonometric functions are functions of an angle used in various branches of mathematics. The primary functions include sine, cosine, and tangent, with secant, cosecant, and cotangent being their reciprocals.
In trigonometric identities and algebra, functions are often interchanged or expressed in terms of each other, depending on what simplifies the situation best. This can include transformations like:
In trigonometric identities and algebra, functions are often interchanged or expressed in terms of each other, depending on what simplifies the situation best. This can include transformations like:
- Sine (\( \sin x \))
- Cosine (\( \cos x \))
- Secant (\( \sec x = \frac{1}{\cos x} \))
Other exercises in this chapter
Problem 90
Explain how the double-angle formulas are derived.
View solution Problem 90
Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\tan x=-5$$
View solution Problem 91
Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\cos ^{2} x-\cos x-1=0$$
View solution Problem 91
How can there be three forms of the double-angle formula for \(\cos 2 \theta ?\)
View solution