Problem 56 - Algebra 2

Question

Verify each identity. $$ \frac{1+\tan \theta}{\tan \theta}=\cot \theta+1 $$

Step-by-Step Solution

Answer

\(\frac{1+\tan \theta}{\tan \theta}=\cot \theta+1\) is indeed a correct identity

Step 1: Substitute the tangent and cotangent

Replace \(tan \theta\) with \(\frac{sin \theta}{cos \theta}\) and \(cot \theta\) with \(\frac{cos \theta}{sin \theta}\). So the equation now becomes: \(\frac{1+\frac{sin \theta}{cos \theta}}{\frac{sin \theta}{cos \theta}}=\frac{cos \theta}{sin \theta}+1\)

Step 2: Simplify the equation

Combine terms to make the equation simpler: \(\frac{cos \theta + sin \theta}{sin \theta} = \frac{cos \theta}{sin \theta} + 1\). The right side of the equation may be rewritten as: \(\frac{cos \theta + sin \theta}{sin \theta}\) as well, hence the equation holds true.

Key Concepts

Understanding Tangent

The tangent function, denoted by \( \tan \theta \), is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine of an angle \( \theta \). In mathematical terms, this is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This means that the tangent function relates directly to the lengths of the sides of a right-angled triangle: opposite over adjacent.
Understanding the tangent function is crucial, especially when dealing with different trigonometric identities and transformations. Here are a few important properties of tangent:

• It is periodic with a period of \( \pi \) radians or 180 degrees.
• It is undefined when \( \cos \theta = 0 \), which occurs at odd multiples of \( \frac{\pi}{2} \).
• The tangent of an angle is positive in the first and third quadrants.

By mastering these properties, students can effectively handle problems involving tangent in various mathematical fields.

Exploring Cotangent

The cotangent function, represented as \( \cot \theta \), is another fundamental trigonometric function which is the reciprocal of tangent. This means that \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
Cotangent has its own unique characteristics that distinguish it from the tangent function:

• It is periodic with a period of \( \pi \) radians, similar to tangent.
• It is undefined when \( \sin \theta = 0 \), occurring at integer multiples of \( \pi \).
• Cotangent is positive in the first and third quadrant, aligning with the sign of tangent.

These properties make it an essential tool in simplifying trigonometric expressions and verifying identities, such as rewriting complex expressions in a simpler cotangent format.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using identities like Pythagorean, quotient, or reciprocal identities. It is a crucial skill to unravel complex problems and make expressions easier to handle. Take the expression \( \frac{1+\tan \theta}{\tan \theta} \), for example:
The first step in simplification is substituting known identities. Here, \( \tan \theta \) is replaced by \( \frac{\sin \theta}{\cos \theta} \). This brings the expression into a form that's easier to simplify.
Next, focus on combining fractions by finding a common denominator or algebraic manipulation. For the given identity, this means working within the framework of the fractions involved to reach a simpler state.
Remember: Simplification often involves strategic rearrangement and substitution to show that two sides of an identity are equal. Keep practicing different problems to become more confident in this skill.

Verifying Identities

Verifying trigonometric identities is about proving that two expressions are equivalent. This can be tackled by transforming one or both sides of an equation using known identities until they appear identical.
Let's consider the identity \( \frac{1+\tan \theta}{\tan \theta} = \cot \theta + 1 \):

• Start by substituting definitions, such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
• Simplify each side using algebraic identities and operations. This may include finding a common denominator or rationalizing the expressions.
• Continue with the transformations until both sides of the equation look the same, confirming the identity.

Verifying identities trains logical thinking and enhances understanding of trigonometric relationships. Through repeated practice, it becomes easier to spot patterns and confidently manage even complicated identities.