Problem 99 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \tan ^{2} 15^{\circ}-\sec ^{2} 15^{\circ}=-1 $$

Step-by-Step Solution

Answer

The statement is True

Step 1: Understand the Problem

The task is to verify the trigonometric statement: \(\tan^{2} 15^{\circ}-\sec^{2} 15^{\circ}=-1\). If it's false, then correct it.

Step 2: Apply the Known Identity

Firstly, recall a trigonometric identity: \(\tan^{2}x +1 = \sec^{2}x\). Rearrange to find: \(\tan^{2}x - \sec^{2}x = -1\). This is the form needed here.

Step 3: Compare the Expression

Now, compare the expression in the problem with the rearranged Pythagorean identity. They match exactly. This shows the given statement is true.

Key Concepts

Pythagorean identities

Pythagorean identities are fundamental relationships between trigonometric functions that derive from the Pythagorean theorem. They help simplify expressions and solve equations involving trigonometric functions. The most common Pythagorean identity is \(\sin^2x + \cos^2x = 1\), reflecting the relationship between the sine and cosine of an angle within a right triangle. Another well-known identity is derived from dividing the fundamental identity by \(\cos^2x\), which gives us \(\tan^2x + 1 = \sec^2x\). This formula relates the tangent and secant functions.
For example, if you need to verify or transform trigonometric expressions, these identities are essential. By rearranging the identity \(\tan^2x + 1 = \sec^2x\), we can also find \(\tan^2x - \sec^2x = -1\). Such rearrangements help solve problems like verifying trigonometric statements.

tangent and secant functions

The tangent and secant functions are two of the six primary trigonometric functions. Understanding them is crucial in trigonometry, especially when working with identities and verifying statements.
The tangent function, represented as \(\tan(x)\), is the ratio of the sine and cosine functions: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). It describes how steep a line is compared to the horizontal axis. It's useful for solving various trigonometric problems, especially in calculus and right triangle trigonometry.
Meanwhile, the secant function is the reciprocal of the cosine function: \(\sec(x) = \frac{1}{\cos(x)}\). This function is less commonly used than others like sine and cosine, but is important in advanced trigonometry. It describes the length of the hypotenuse in a unit circle framework for a given angle, helping to verify complex trigonometric identities.
Both functions appear in Pythagorean identities, highlighting their relationship and showing how they complement each other in equations like \(\tan^2x + 1 = \sec^2x\).

verifying trigonometric statements

Verifying trigonometric statements involves checking the accuracy of equations or expressions that use trigonometric identities. The process requires a strong understanding of these identities and an ability to manipulate algebraic expressions.
When tasked with verification, such as checking if a statement like \(\tan^2 15^\circ - \sec^2 15^\circ = -1\) is true, applying known identities is essential. Here, you would use the Pythagorean identity \(\tan^2x + 1 = \sec^2x\), which, when rearranged, matches the statement exactly: \(\tan^2x - \sec^2x = -1\).
Steps for verification typically include:

• Identify relevant identities connected to the trigonometric functions in the statement.
• Reorganize the identities to match the form of the given expression.
• Compare the original and transformed expressions to confirm their equivalence or discrepancy.

Through these steps, you can determine if a statement is true or false and make any necessary corrections. This practice enhances your understanding of both the identities and the functions involved, making it a vital skill in trigonometry.