Problem 3 - Precalculus

Question

Use the Even / Odd Identities to verify the identity. Assume all quantities are defined. $$ \tan \left(-t^{2}+1\right)=-\tan \left(t^{2}-1\right) $$

Step-by-Step Solution

Answer

The identity is verified using the odd function property of tangent: \( \tan(-t^2+1) = -\tan(t^2-1) \).

Step 1: Recognize the Identity

Start by recognizing that the given identity involves an even/odd identity for the tangent function. Tangent is an odd function, which means \( \tan(-x) = -\tan(x) \).

Step 2: Apply the Odd Identity

Apply the odd identity to the expression on the left side of the equation: \( \tan(-t^2 + 1) = \tan(-(t^2 - 1)) \).

Step 3: Simplify the Expression

Using the property of odd functions, we can simplify \( \tan(-(t^2 - 1)) \) to \(-\tan(t^2 - 1) \). This follows since \( \tan(-x) = -\tan(x) \).

Step 4: Compare Both Sides

Now both sides of the equation have been simplified to the same expression, \(-\tan(t^2 - 1) \). Thus, the given identity is verified.

Key Concepts

Even and Odd Functions

In mathematics, functions can often be classified into two categories: even and odd functions. Whether a function is even or odd primarily depends on its symmetry properties. This classification helps in simplifying expressions and verifying identities.

• Even Functions: These functions are symmetrically balanced around the y-axis. The defining property is that for every x in the domain, the function satisfies the relation: \( f(-x) = f(x) \).
• Odd Functions: These possess origin symmetry. This means that rotating the graph 180 degrees about the origin yields the same graph. For odd functions, the defining equation is: \( f(-x) = -f(x) \). These are key for understanding how functions behave under sign changes.

The tangent function, \( \tan(x) \), is an excellent example of an odd function. If you calculate \( \tan(-x) \), you’ll find it equals \(-\tan(x) \). This property helps simplify trigonometric identities extensively.

Tangent Function Properties

The tangent function, noted as \( \tan(x) \), is one of the primary trigonometric functions. Understanding its properties can be crucial when dealing with trigonometric identities. Consider the following:

• Periodicity: The tangent function is periodic with period \( \pi \), meaning \( \tan(x + \pi) = \tan(x) \).
• Odd Function: As discussed earlier, \( \tan(x) \) satisfies \( \tan(-x) = -\tan(x) \). This is useful for understanding how transformations affect the graph.
• Zeros and Asymptotes: \( \tan(x) \) equals zero at integer multiples of \( \pi \), specifically \( x = n \pi \), where n is an integer, and has vertical asymptotes at \( x = (2n+1)\frac{\pi}{2} \).

When verifying or manipulating expressions involving \( \tan(x) \), remember these properties to quickly check or transform identities and equations.

Verifying Identities

Verifying trigonometric identities involves showing that two expressions are equivalent for all values of the variable within a specified domain. Utilizing the properties of functions - whether they are even, odd, or periodic - can simplify this process.

• Recognize the Type: If an identity involves functions you recognize as even or odd, recall those properties immediately. For instance, the identity \( \tan(-x) = -\tan(x) \) quickly leads to simplifications.
• Use Algebraic Manipulations: Often these will involve basic algebra such as factoring, expanding, or combining fractions.
• Check Both Sides: After manipulating, ensure both sides of the identity match using properties and known formulas.

In our exercise, the identity \( \tan(-t^{2}+1)=-\tan(t^{2}-1) \) was verified by recognizing the odd function property of tangent. Both sides were reduced to the same expression, proving the identity's correctness. Developing intuition for these properties enriches problem-solving efficiency.