Problem 54 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{1-\cos \alpha}{1+\cos \alpha}=\frac{\sec \alpha-1}{\sec \alpha+1} $$

Step-by-Step Solution

Answer

The identity is verified by substituting \( \sec \alpha = \frac{1}{\cos \alpha} \) and simplifying the right-hand side to match the left-hand side.

Step 1: Analyze the Given Identity

The problem asks to verify the identity \( \frac{1-\cos \alpha}{1+\cos \alpha}=\frac{\sec \alpha-1}{\sec \alpha+1} \). This means we have to show that both sides are equal. We'll start by expressing all trigonometric functions in terms of sine and cosine.

Step 2: Rewrite \( \sec \alpha \)

Remember that \( \sec \alpha = \frac{1}{\cos \alpha} \). Substitute this into the right-hand side to get:\[ \frac{\frac{1}{\cos \alpha} - 1}{\frac{1}{\cos \alpha} + 1} \] This simplifies to \( \frac{\frac{1-\cos\alpha}{\cos\alpha}}{\frac{1+\cos\alpha}{\cos\alpha}} \).

Step 3: Simplify the Right-Hand Side

Divide the fractions in the right-hand side:\[ \frac{1-\cos\alpha}{1+\cos\alpha} \]. Notice this is now identical to the left-hand side, showing that both expressions are indeed equal.

Key Concepts

Trigonometric Functions

Trigonometric functions are the foundation of trigonometry, dealing with angles and the relationships between them. These functions include sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), and their reciprocals like secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)). In our exercise, both cosine and secant functions are utilized.

Remember, \(\cos \alpha\) is the adjacent over hypotenuse in a right triangle, while \(\sec \alpha\) is the reciprocal of cosine, meaning \(\sec \alpha = \frac{1}{\cos \alpha}\). Converting \(\sec \alpha\) into cosine helps simplify expressions and solve identities.

When working with identities, it is key to express all functions in terms of basic sine and cosine. This strategy simplifies the expressions since all other functions can be derived from these baselines.

Verifying Identities

Verifying trigonometric identities involves proving that two expressions are equal. It's like working through a puzzle where each piece must perfectly match the other.

Here are steps to verify an identity effectively:

• Start by simplifying each side of the identity separately, converting everything to sine and cosine where possible.
• Look for common trigonometric identities such as Pythagorean identities or angle sum identities that can simplify expressions further.
• Work step-by-step, keeping each transformation valid and logical.
• The goal is to manipulate one or both expressions until they appear identical.

In our exercise, we took the problem expression and substituted \(\sec \alpha\) as \(\frac{1}{\cos \alpha}\) to help reach the solution. By simplifying the right-hand expression, we demonstrated that it equaled the left-hand side, thus proving the identity.

Simplifying Fractions

Simplifying fractions in trigonometry often involves trickier steps due to the presence of functions like sine and cosine.

When dealing with complex fractions, as seen in our exercise, a strategic approach makes all the difference:

• Factorize the terms if possible; factorizing sometimes simplifies the global structure of an expression.
• Cancel out common factors across numerator and denominator to reduce complexity.
• Use common denominators or multiply by conjugates when necessary to achieve a simpler form.

In the given problem, substituting \(\sec \alpha\) with \(\frac{1}{\cos \alpha}\) led to nested fractions. Dividing these fractions was key to simplifying them to a match the left-hand side of the identity.

This approach underscores the importance of breaking down each component and simplifying strategically when verifying identities.