Problem 47 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \sqrt{\frac{1+\sin \theta}{1-\sin \theta}}=\frac{1+\sin \theta}{|\cos \theta|} $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\frac{1+\sin \theta}{|\cos \theta|}\).

Step 1: Simplify the Left Side

Start with the left side of the given identity: \( \sqrt{\frac{1+\sin \theta}{1-\sin \theta}} \). To simplify, multiply both the numerator and the denominator by the conjugate of the denominator, \(1+\sin \theta\), to rationalize it: \( \sqrt{\frac{(1+\sin \theta)^2}{(1-\sin \theta)(1+\sin \theta)}} \).

Step 2: Perform Algebraic Operations

Next, in the denominator, apply the difference of squares formula: \((1-\sin \theta)(1+\sin \theta) = 1^2 - \sin^2\theta = \cos^2\theta\). Now, the expression simplifies to \(\sqrt{\frac{(1+\sin \theta)^2}{\cos^2\theta}}\).

Step 3: Simplify the Square Root Expression

Apply the square root to both the numerator and the denominator: \(\frac{\sqrt{(1+\sin \theta)^2}}{\sqrt{\cos^2\theta}}\). Simplify further: \(\sqrt{(1+\sin \theta)^2} = |1+\sin \theta|\) and \(\sqrt{\cos^2\theta} = |\cos \theta|\).

Step 4: Verify the Identity

Since we simplified the left side to \(\frac{|1+\sin \theta|}{|\cos \theta|}\), and the right side is given as \(\frac{1+\sin \theta}{|\cos \theta|}\), notice that \(1+\sin \theta\) is generally positive. Thus, \(|1+\sin \theta| = 1+\sin \theta\). Now both sides match: \(\frac{1+\sin \theta}{|\cos \theta|}\). Thus, the given identity is verified.

Key Concepts

Rationalizing Expressions

Rationalizing expressions is a technique often used in algebra and trigonometry to simplify complex fractions by eliminating any square roots in the denominator. In the context of trigonometry, this process can make expressions easier to work with, especially when proving identities or simplifying equations. To rationalize a denominator that consists of a binomial with a square root, you multiply both the numerator and denominator by the conjugate of the denominator.

The conjugate of a binomial like \(1 - \sin \theta\) is \(1 + \sin \theta\). When you multiply an expression by its conjugate, you take advantage of the difference of squares: \((a-b)(a+b) = a^2 - b^2\). This operation eliminates the square root, resulting in just numbers or simpler expressions in the denominator.

By multiplying the numerator and denominator by \(1 + \sin \theta\) in the original problem, you clear out the root from the denominator. You end up simplifying the expression into a more manageable form, making it easier to verify or transform the identity in question.

Difference of Squares

The difference of squares is a powerful algebraic identity, used to simplify expressions and solve equations. This technique states that the product of the sum and the difference of two terms is equal to the difference of their squares. Mathematically, this is written as \((a-b)(a+b) = a^2 - b^2\).

In our trigonometric identity exercise, the difference of squares allows us to transform complex trigonometric terms into simpler forms. Specifically, the expression \((1-\sin \theta)(1+\sin \theta)\) becomes \(1^2 - \sin^2\theta\) or \(\cos^2\theta\).

Using this transformation unravels the complexity of the expression and emphasizes the connection between different parts of trigonometry. Recognizing and applying the difference of squares is essential in simplifying and verifying trigonometric identities, allowing you to see the equivalence between different expressions more clearly.

Absolute Value in Trigonometry

Understanding absolute values in trigonometry is crucial when dealing with expressions that have square roots. The absolute value of a number is its distance from zero on the number line, disregarding any negative sign. In trigonometry, we encounter absolute values quite frequently, particularly when dealing with root expressions.

In the solution given for the trigonometric identity \(\sqrt{(1+\sin \theta)^2} = |1+\sin \theta|\) and \(\sqrt{\cos^2\theta} = |\cos \theta|\), absolute values help ensure that the expressions inside the square root remain non-negative. This is because the square root of a square gives the absolute value of the original term.

Moreover, considering the absolute value makes the identity valid for all possible input values of \(\theta\), as trigonometric functions like sine and cosine can take on negative values depending on the angle's quadrant. This ensures that our mathematical arguments hold universally, reinforcing the identity as true across all conditions.