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Magazine Chapter 4: Problem 21
Reduce the given expression to a single trigonometric function. $$ \frac{1}{1+\sin t}+\frac{1}{1-\sin t} $$
Short Answer
Expert verified
The expression simplifies to \(2 \sec^2 t\).
Step by step solution
01
Find a Common Denominator
To simplify the expression \( \frac{1}{1+\sin t} + \frac{1}{1-\sin t} \), we need a common denominator. The common denominator of \( \frac{1}{1+\sin t} \) and \( \frac{1}{1-\sin t} \) is \((1+\sin t)(1-\sin t)\).
02
Express Each Fraction with the Common Denominator
Rewrite each fraction so that it has the common denominator:\[ \frac{1}{1+\sin t} = \frac{1-\sin t}{(1+\sin t)(1-\sin t)} \]\[ \frac{1}{1-\sin t} = \frac{1+\sin t}{(1+\sin t)(1-\sin t)} \]
03
Add the Fractions
Combine the fractions into a single expression:\[ \frac{1-\sin t + 1+\sin t}{(1+\sin t)(1-\sin t)} \]
04
Simplify the Numerator
Simplify the expression in the numerator:\[ 1 - \sin t + 1 + \sin t = 2 \]
05
Simplify the Denominator
The denominator \((1+\sin t)(1-\sin t)\) simplifies using the difference of squares: \[ (1+\sin t)(1-\sin t) = 1 - \sin^2 t \]
06
Apply Trigonometric Identity
Use the Pythagorean identity \(1 - \sin^2 t = \cos^2 t\), thus the expression is now:\[ \frac{2}{\cos^2 t} \]
07
Express as a Single Trigonometric Function
The expression \( \frac{2}{\cos^2 t} \) can be written as \(2 \sec^2 t\). Hence, the expression reduces to a single trigonometric function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
When dealing with expressions like \((1+\sin t)(1-\sin t)\), we can utilize a neat mathematical trick called the "Difference of Squares." This is an algebraic identity that states: \[(a + b)(a - b) = a^2 - b^2\]Applying this identity to \((1+\sin t)(1-\sin t)\), we set \(a = 1\) and \(b = \sin t\). Thus, \[(1+\sin t)(1-\sin t) = 1^2 - (\sin t)^2 = 1 - \sin^2 t\]This transformation is crucial in simplifying many trigonometric expressions, as you often encounter terms that fit this pattern.
- It is commonly used in calculus and algebraic manipulations.
- It simplifies the process of factoring and expanding expressions.
- It also plays a vital role in trigonometric identities.
Common Denominator
Finding a common denominator is an essential step when adding or subtracting fractions. In our original problem, we needed one to combine \(\frac{1}{1+\sin t} + \frac{1}{1-\sin t}\).A common denominator ensures that fractions can be easily added or subtracted.For the fractions given, the common denominator is \((1+\sin t)(1-\sin t)\). This allows each fraction to be rewritten as:
- \(\frac{1-\sin t}{(1+\sin t)(1-\sin t)}\)
- \(\frac{1+\sin t}{(1+\sin t)(1-\sin t)}\)
Pythagorean Identity
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables for which both sides of the equation are defined. A particularly important identity is the Pythagorean Identity. It states:\[\sin^2 t + \cos^2 t = 1\]From this identity, we can express \(1 - \sin^2 t\) as \(\cos^2 t\), which simplifies our denominator: \(1 - \sin^2 t = \cos^2 t\).
- This identity is fundamental in trigonometry.
- It links sine and cosine, allowing for transformation and simplification of expressions.
- It is used when converting between expressions involving squared trigonometric functions.
Secant Function
The secant function is one of the basic trigonometric functions, and it is defined in terms of the cosine function:\[\sec t = \frac{1}{\cos t}\].In the context of our problem, the reduced expression was\(\frac{2}{\cos^2 t}\).Recognizing that \(\frac{1}{\cos^2 t}\) is \(\sec^2 t\), the final result becomes:\[2 \sec^2 t\].
- The secant function is specifically useful in problems involving reciprocal trigonometric expressions.
- It is important in calculus for derivatives and integrals involving trigonometric functions.
- Understanding secant can simplify complex trigonometric solutions.
