Question

Show algebraically that \(\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=2 \sec ^{2} \theta\) is an identity.

Short answer

It is shown that \[\frac{1}{1+\text{sin}\theta}+\frac{1}{1-\text{sin}\theta} = 2\text{sec}^2\theta.\]

Step-by-step solution

- Express in a common denominator

Rewrite the expression \(\frac{1}{1+\text{sin}\theta} + \frac{1}{1-\text{sin}\theta}\) with a common denominator. The common denominator for these fractions is \( (1+\text{sin}\theta)(1-\text{sin}\theta) \).

- Combine under the common denominator

Combine the fractions: \[ \frac{(1-\text{sin}\theta)+(1+\text{sin}\theta)}{(1+\text{sin}\theta)(1-\text{sin}\theta)}. \]

- Simplify the numerator

Simplify the numerator: \[(1 - \text{sin}\theta) + (1 + \text{sin}\theta) = 2.\]

- Simplify the denominator

Simplify the denominator: \[(1+\text{sin}\theta)(1-\text{sin}\theta) = 1 - \text{sin}^2\theta \].

- Use Pythagorean identity

Recall the Pythagorean identity \( 1 - \text{sin}^2\theta = \text{cos}^2\theta \). Substituting this in the denominator, we get \[\frac{2}{\text{cos}^2\theta}.\]

- Result in terms of secant

Recognize that \( \text{sec}\theta = \frac{1}{\text{cos}\theta} \). Thus, \[\frac{2}{\text{cos}^2\theta} = 2\text{sec}^2\theta\].

- Conclude the identity proof

Thus, it can be concluded that \[\frac{1}{1+\text{sin}\theta}+\frac{1}{1-\text{sin}\theta} = 2\text{sec}^2\theta \].

Key concepts

Common Denominator

When faced with adding or subtracting fractions, finding a common denominator ensures the process goes smoothly. To start, identify the denominators of the fractions you are working with. For our exercise, the fractions are \(\frac{1}{1+\text{sin}\theta}\) and \(\frac{1}{1-\text{sin}\theta}\). The common denominator is the product of the two different denominators: \((1+\text{sin}\theta)(1-\text{sin}\theta)\). This step is crucial for combining the fractions. By rewriting them with a common denominator, we are able to add or subtract them accordingly.

• The product rule of denominators helps in achieving a common base.
• Having a common denominator simplifies the overall expression for the next steps.

Always look for the least common multiple (LCM) of the denominators when dealing with more complex fractions.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that connects the sine and cosine functions. The identity is: \[ 1 - \text{sin}^2\theta = \text{cos}^2\theta \]. This comes in handy when simplifying trigonometric expressions. In our exercise, after expressing the fractions with a common denominator, we noticed that the denominator simplified to \1 - \text{sin}^2\theta\. By applying the Pythagorean identity, we replaced this with \ \text{cos}^2\theta \.
This identity not only simplifies calculations but also helps in transforming trigonometric expressions into more familiar or workable forms.

• The Pythagorean identity simplifies the proofs involving sin and cos.
• It helps in transforming products of trigonometric functions.

Remember, this identity is always valid for any angle \theta\.

Secant Function

The secant function, denoted as \ \text{sec} \theta \, is the reciprocal of the cosine function. Mathematically, it is defined as: \[ \text{sec}\theta = \frac{1}{\text{cos}\theta} \]. In our exercise, once we simplified the expression to \ \frac{2}{\text{cos}^2\theta} \, recognizing the secant function was key to tying everything together. By knowing that \ \text{sec}\theta = \frac{1}{\text{cos}\theta} \, it follows that \ \frac{2}{\text{cos}^2\theta} \ simplifies to \ 2\text{sec}^2\theta \.

• The secant function is often used in identities and simplifications.
• It is a critical function for hyperbolic and inverse trigonometric relationships.

Recognizing \ \text{sec}\theta \ and its properties is essential in transforming and evaluating trigonometric expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging, simplifying and solving expressions using algebraic rules. For our problem, it included:

• Decomposing and distributing terms.
• Combining like terms.
• Using known identities to simplify expressions.

In the exercise, combining fractions required a common denominator. We then simplified the numerator and the denominator using algebraic rules. Finally, recognizing and applying trigonometric identities allowed us to prove the given identity.

Mastering algebraic manipulation comes down to practice. Step by step, one can simplify and transform complex expressions into forms where known identities or values can be easily applied.

• Always proceed methodically.
• Keep track of each step.
• Verify by back substituting.

The foundation of proving any identity lies in solid algebraic manipulation skills. This makes the process intuitive and systematic.