Problem 26 - Technical Mathematics with Calculus

Question

Simplify. $$\frac{1}{1+\sin x}+\frac{1}{1-\sin x}$$

Step-by-Step Solution

Answer

\(\frac{2}{\cos^2 x}\)

Step 1: Combine the fractions

Find a common denominator for the two fractions, which is \(1 + \sin x)(1 - \sin x)\). Multiply the numerators of each fraction by the denominator of the other fraction to combine them into a single fraction.

Step 2: Apply the difference of squares formula

Recognize that the common denominator \(1 + \sin x)(1 - \sin x)\) is a difference of squares, which can be simplified to \(1 - \sin^2 x\), which is equivalent to \(\cos^2 x\) by the Pythagorean identity.

Step 3: Simplify the numerator

Multiply the numerators from Step 1 and add them together to find the combined numerator of the single fraction.

Step 4: Simplify the final expression

Divide the simplified numerator by the simplified denominator to get the final simplified expression.

Key Concepts

Common Denominator

When adding or subtracting fractions with different denominators, finding a common denominator is essential. This ensures that the fractions are represented in parts of the same size, making it possible to combine them. In the case of trigonometric expressions, as we have with \(\frac{1}{1+\sin x}+\frac{1}{1-\sin x}\), the common denominator will often involve trigonometric functions.

The common denominator can be found by multiplying the denominators together, resulting in \( (1 + \sin x)(1 - \sin x)\). This allows us to combine the fractions into a single fraction, which sets the stage for further simplification using other trigonometric identities.

Difference of Squares

The difference of squares is an algebraic pattern that takes the form \(a^2 - b^2\) and can be factored into \( (a - b)(a + b) \). This property is incredibly useful in simplifying expressions, particularly when dealing with trigonometric functions.

In our exercise, the common denominator derived in the first step, \( (1 + \sin x)(1 - \sin x)\), is a clear representation of the difference of squares pattern, with \(a = 1\) and \(b = \sin x\). By applying this property, the denominator simplifies to \(1^2 - (\sin x)^2\), or equivalently \(1 - \sin^2 x\), which sets up the opportunity to employ trigonometric identities to further simplify the expression.

Pythagorean Identity

The Pythagorean identity in trigonometry is an equation that relates the squares of the sine and cosine of an angle. Specifically, it states that \(\sin^2 x + \cos^2 x = 1\). This identity is derived from the Pythagorean theorem and is a fundamental relationship in trigonometric simplification.

By rearranging the identity, we can express \(\sin^2 x\) as \(1 - \cos^2 x\) or \(\cos^2 x\) as \(1 - \sin^2 x\). In the context of our exercise, recognizing the expression \(1 - \sin^2 x\) as \(\cos^2 x\) allows us to replace the denominator with this simpler, equivalent expression. Understanding and applying the Pythagorean identity is crucial for simplifying complex trigonometric expressions into a form that is more easily managed and understood.

Trigonometric Simplification

Trigonometric simplification refers to the process of reducing a trigonometric expression to its simplest form. This often involves using identities, such as the Pythagorean identity, to eliminate functions or terms and make the expression more straightforward.

It is not just about making an expression look neater; simplification can reveal important relationships and make further calculations or integrations more feasible. In our solved problem, once we've combined the fractions using a common denominator and applied the difference of squares, the Pythagorean identity transforms the complex-looking fraction into an expression involving only one trigonometric function, \(\cos\). With this, we can simplify the numerator accordingly and arrive at a much simpler form, illustrating the power of trigonometric simplification in solving and understanding trigonometric equations.