Problem 26 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{1+\tan t}{\tan t}=\cot t+\sec ^{2} t-\tan ^{2} t $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \cot t + 1 \).

Step 1: Simplify the Left Side

Start by simplifying the left side of the identity, \( \frac{1+\tan t}{\tan t} \). Split this fraction into two separate fractions: \( \frac{1}{\tan t} + \frac{\tan t}{\tan t} \). Notice that \( \frac{1}{\tan t} \) is equivalent to \( \cot t \) and \( \frac{\tan t}{\tan t} \) is equal to 1. Therefore, the left side simplifies to \( \cot t + 1 \).

Step 2: Simplify the Right Side

Now, simplify the right side of the identity, \( \cot t + \sec^{2} t - \tan^{2} t \). Use the trigonometric identity \( \sec^{2} t = 1 + \tan^{2} t \). Substituting this identity, the expression becomes \( \cot t + (1 + \tan^{2} t) - \tan^{2} t \). Simplify further to get \( \cot t + 1 \) as the \( \tan^2 t \) terms cancel each other.

Step 3: Verify the Identity

Compare the simplified expressions from Step 1 and Step 2. Both simplify to \( \cot t + 1 \). Therefore, the original identity \( \frac{1+\tan t}{\tan t} = \cot t + \sec^2 t - \tan^2 t \) is verified as correct.

Key Concepts

Simplifying expressions

Simplifying expressions in trigonometry involves reducing complex fractions or equations to more manageable forms. In the exercise we are working with, the goal is to simplify both sides of the given trigonometric identity and prove that they are equal. Starting with the left side, we have the expression \( \frac{1+\tan t}{\tan t} \). This expression can be split into two separate parts, \( \frac{1}{\tan t} \) and \( \frac{\tan t}{\tan t} \).

• \( \frac{1}{\tan t} \) simplifies to \( \cot t \) because cotangent is the reciprocal of tangent.
• \( \frac{\tan t}{\tan t} \) simplifies to 1 since any non-zero number divided by itself is 1.

With these simplifications, the left side becomes \( \cot t + 1 \). This process of breaking down and rearranging terms by using known identities or algebraic rules is crucial for managing more complex trigonometric expressions. It helps in identifying equal parts and hence proving the identity.

Verifying identities

Verifying trigonometric identities involves showing that two expressions are equivalent using algebraic manipulations and known trigonometric identities. For the identity \( \frac{1+\tan t}{\tan t}=\cot t+\sec ^{2} t-\tan ^{2} t \), both sides need to be simplified and shown to equate to the same expression.

• After simplifying the left side to \( \cot t + 1 \), as described earlier, we move on to the right side: \( \cot t + \sec^{2} t - \tan^{2} t \).
• Using the identity \( \sec^{2} t = 1 + \tan^{2} t \), substitute and simplify this right side: \( \cot t + (1 + \tan^{2} t) - \tan^{2} t \).

The \( \tan^2 t \) terms cancel each other, leaving \( \cot t + 1 \). Successfully simplifying both sides to the same expression, \( \cot t + 1 \), confirms that the original identity is indeed valid. Verifying identities requires practice and the ability to recognize how different trigonometric functions relate to one another.

Trigonometric functions

Trigonometric functions are fundamental tools in mathematics that relate the angles of triangles to their side lengths. For example, in our exercise, functions such as tangent (\( \tan t \)), cotangent (\( \cot t \)), and secant (\( \sec t \)) are crucial. Each function has a specific role and relationship:

• \( \tan t \) is the ratio of the opposite side to the adjacent side in a right triangle.
• \( \cot t = \frac{1}{\tan t} \) or opposite of \( \tan t \), is the reciprocal function.
• \( \sec t = \frac{1}{\cos t} \) is the reciprocal of cosine, meaning it is defined as \( 1/\cos t \).

Understanding these relationships can simplify the process of working with identities. These functions allow us to express complex relationships in simpler terms, making it easier to verify identities or solve equations. Trigonometric identities, such as \( \sec^{2} t = 1 + \tan^{2} t \), are derived from these basic functions and help unite complex expressions under a single form.