Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\tan \alpha}{1+\sec \alpha}+\frac{1+\sec \alpha}{\tan \alpha}=2 \csc \alpha $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \(2 \csc \alpha\).

Step 1: Simplify the LHS

Write the left-hand side (LHS) of the expression as a single term. Start with:\[ \frac{\tan \alpha}{1+\sec \alpha} + \frac{1+\sec \alpha}{\tan \alpha} \]Find a common denominator, which is \(\tan \alpha (1+\sec \alpha)\). Combine the terms:\[ \frac{\tan^2 \alpha + (1+\sec \alpha)^2}{\tan \alpha (1+\sec \alpha)} \]

Step 2: Apply Trigonometric Identities

Replace \(\tan \alpha\) with \(\frac{\sin \alpha}{\cos \alpha}\) and \(\sec \alpha\) with \(\frac{1}{\cos \alpha}\). This transforms the expression into terms of sine and cosine. The substitution makes it easier to further simplify.\[ \frac{\left( \frac{\sin^2 \alpha}{\cos^2 \alpha} + \left(1 + \frac{1}{\cos \alpha}\right)^2 \right) \cos \alpha \cos \alpha}{\sin \alpha (\cos \alpha + 1)} \]

Step 3: Simplify the Expression Using Trigonometric Identities

Simplify using identities like \( \sin^2 \alpha + \cos^2 \alpha = 1 \) and cancel appropriate terms where possible. Continue to simplify until the expression takes a more recognizable form:\[ \frac{\sin^2 \alpha + 1 + 2\sec \alpha + \sec^2 \alpha}{\sin \alpha (1+\sec \alpha)} \]

Step 4: Further Simplify and Compare with RHS

Recognize that \(1 + \sec^2 \alpha = \sec \alpha + \tan^2 \alpha\), allowing cancellations in the expression, and get:\[ \frac{2}{\sin \alpha} \]Notice that this expression equals \(2 \csc \alpha\), proving the original identity true.

Key Concepts

Trig Functions

Understanding trigonometric functions is essential for delving into trigonometric identities. These functions, often abbreviated as trig functions, are fundamental in mathematics, especially in the study of right triangles and circular motion. The primary trig functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). These relate to the angles and sides of a right triangle and to positions on the unit circle.

• Sine (\( \sin \theta \)): This function represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos \theta \)): This is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): Defined as the ratio of the opposite side to the adjacent side, it can also be expressed as \( \frac{\sin \theta}{\cos \theta} \).

In addition to these primary functions, there are reciprocal functions such as secant (\( \sec \theta \)), cosecant (\( \csc \theta \)), and cotangent (\( \cot \theta \)), which help in expanding the scope of trigonometric calculations. For instance, \( \sec \theta \) is the reciprocal of cosine, and \( \csc \theta \) is the reciprocal of sine. Understanding these allows for more complex identity verifications and problem-solving in trigonometry.

Proof Verification

Verification of mathematical identities is a crucial process. It involves comparing left-hand side (LHS) and right-hand side (RHS) expressions to establish equality. In trigonometrics, proof verification helps confirm the relationships between different trig functions.
To verify an identity, begin by simplifying the more complex side of the equation, often using known trigonometric identities. Trigonometric identities like \( \sin^2 \alpha + \cos^2 \alpha = 1 \) are fundamental, as they help to transform and reduce expressions.

• Simplifying one side often involves converting all terms into sine and cosine, as these are the most basic trig functions.
• Look for opportunities to factor or use substitution to match formats between sides.

For example, in verifying an identity that involves \( \tan \alpha \) and \( \sec \alpha \), substituting \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \sec \alpha = \frac{1}{\cos \alpha} \) can simplify the process significantly. By reaching an identical form on both sides, you successfully complete the verification, proving the identity is true.

Sine and Cosine

Sine and cosine functions are the building blocks of trigonometry. These functions allow us to explore and prove various trigonometric identities.

• Sine ( \( \sin \alpha \) ): This function gives a measure of vertical distance or height on the unit circle.
• Cosine ( \( \cos \alpha \) ): This represents the horizontal distance or length on the unit circle.

The relationship between sine and cosine is epitomized by the Pythagorean identity: \[\sin^2 \alpha + \cos^2 \alpha = 1\]. This powerful identity is used extensively in simplifying and manipulating equations. In identity proofs, replacing tangent and secant with their definitions in terms of sine and cosine simplifies complex expressions.
To prove the original example identity\[\frac{\tan \alpha}{1+\sec \alpha}+\frac{1+\sec \alpha}{\tan \alpha}=2 \csc \alpha\], using \( \sin \theta \) and \( \cos \theta \) aids in breaking down the expression into manageable parts. Understanding these functions grants the capability to tackle intricate mathematical problems with clarity.