Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$(1+\tan s)^{2}-2 \tan s$$

Step-by-Step Solution

Answer

The expression simplifies to \(\sec^2 s\).

Step 1: Use Algebraic Expansion

Expand the expression \((1+\tan s)^{2}\) by using the formula \(a^2 = (a+b)^2 = a^2 + 2ab + b^2\), where \(a = 1\) and \(b = \tan s\). So, the expansion is \(1^2 + 2 \cdot 1 \cdot \tan s + (\tan s)^2 = 1 + 2\tan s + \tan^2 s\).

Step 2: Substitute Back into Original Expression

Substitute the expanded expression \(1 + 2\tan s + \tan^2 s\) back into the original expression: \((1+\tan s)^{2}-2 \tan s\). This gives us \(1 + 2\tan s + \tan^2 s - 2\tan s\).

Step 3: Simplify

Combine like terms in the expression. \(1 + 2\tan s + \tan^2 s - 2\tan s = 1 + \tan^2 s\). The \(2\tan s\) terms cancel each other out.

Step 4: Use Trigonometric Identity

Recognize that \(1 + \tan^2 s\) is a known trigonometric identity, which is \(\sec^2 s\). Thus, the expression simplifies to \(\sec^2 s\).

Key Concepts

Algebraic Expansion

Algebraic expansion is a technique used to simplify expressions that involve polynomials or other similar expressions. It's like unwrapping a package to see what's inside the brackets. In this exercise, we have \((1+\tan s)^{2}\), which is a perfect square trinomial.
To expand this expression, we use the formula \((a+b)^2 = a^2 + 2ab + b^2\).

• Here, \(a\) is \(1\) and \(b\) is \(\tan s\).
• Substituting these values, we get \(1^2 + 2 \cdot 1 \cdot \tan s + (\tan s)^2\).
• This simplifies to \(1 + 2\tan s + \tan^2 s\).

Algebraic expansion helps us break down complicated expressions into manageable terms. This process lays the groundwork for the next steps, such as simplification and further application of identities.

Simplification

Simplification involves reducing an expression to its simplest form. Once we've expanded our expression using algebraic techniques, the next step is to simplify it. We take the expanded expression: \(1 + 2\tan s + \tan^2 s - 2\tan s\).
Notice that there are like terms present in the expression which are \(2\tan s\) and \(-2\tan s\).

• These terms cancel each other out since they add up to zero.
• What remains is \(1 + \tan^2 s\).

Simplification ensures that the expression is as compact as possible, making it easier to understand and work with in further calculations, like applying trigonometric identities.

Trigonometric Functions

Trigonometric functions relate angles to side lengths in a right triangle. They are fundamental to understanding trigonometric identities. In this particular exercise, an important identity arises: \(1 + \tan^2 s = \sec^2 s\).
This identity is one of the Pythagorean identities, and it connects the tangent function to the secant function.

• \(\tan s\) is the ratio of the opposite side to the adjacent side of a triangle.
• \(\sec s\) is the reciprocal of the cosine function, which is the ratio of the hypotenuse to the adjacent side.

Recognizing these identities allows us to transform and simplify trigonometric expressions more easily. Thus, at the end of this exercise, the expression \(1 + \tan^2 s\) simplifies neatly to \(\sec^2 s\), showcasing the power and utility of trigonometric functions in algebra.