Problem 7 - Algebra and Trigonometry

Question

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \tan ^{2} x-\sec ^{2} x $$

Step-by-Step Solution

Answer

The simplified expression is \(-1\).

Step 1: Express in terms of sine and cosine

Start by converting the trigonometric functions to sine and cosine. Recall the identities: \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). Thus, \(\tan^2 x = \left(\frac{\sin x}{\cos x}\right)^2\) and \(\sec^2 x = \left(\frac{1}{\cos x}\right)^2\).

Step 2: Substitute the expressions

Now that we have the expressions, substitute them into the original expression: \[ \tan^2 x - \sec^2 x = \left(\frac{\sin^2 x}{\cos^2 x}\right) - \left(\frac{1}{\cos^2 x}\right) \]

Step 3: Combine over a common denominator

The common denominator for the terms is \(\cos^2 x\). Write the expression as: \[ \frac{\sin^2 x - 1}{\cos^2 x} \]

Step 4: Use Pythagorean identity

Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), which implies \( \sin^2 x - 1 = -\cos^2 x \). Substitute: \[ \frac{-\cos^2 x}{\cos^2 x} \]

Step 5: Simplify the expression

The expression \(\frac{-\cos^2 x}{\cos^2 x}\) simplifies to \(-1\).

Key Concepts

Sine and Cosine

Sine (\( \sin \)) and cosine (\( \cos \)) are fundamental concepts in trigonometry. They represent the ratio of the sides of a right-angled triangle relative to an angle.
Sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine is the ratio of the length of the adjacent side to the hypotenuse.

• Mathematically, for an angle \( \theta \), these are expressed as:
• Sine: \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)
• Cosine: \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

Using these definitions, every trigonometric function can be expressed in terms of sine and cosine.
For example, the tangent function is related to sine and cosine:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]Understanding these relationships allows us to convert between different trigonometric functions.
Such conversions are often required in solving trigonometric equations and proving identities.

Tangent and Secant

Tangent (\( \tan \)) and secant (\( \sec \)) are derived functions from sine and cosine.
The tangent of an angle \( \theta \) is calculated as the ratio of the sine of the angle to the cosine of the angle.
It represents the slope of the side opposite to the angle in a right triangle divided by the adjacent side.

• The formula for tangent is: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Secant, on the other hand, is the reciprocal of cosine.
It relates to the length of the hypotenuse over the adjacent side in a right triangle:

• Secant is given by: \( \sec \theta = \frac{1}{\cos \theta} \)

Both tangent and secant are essential for expressing and simplifying complex trigonometric expressions.
In the exercise above, expressing both \( \tan^2 x \) and \( \sec^2 x \) using sine and cosine was the first step in the simplification.
This allowed for a common denominator to be applied, simplifying the overall expression effectively.

Pythagorean Identity

The Pythagorean identity is one of the core identities in trigonometry.
It is derived from the Pythagorean theorem applied to a unit circle.
This identity states that the square of the sine of an angle plus the square of the cosine of the same angle always equals one.

• The identity is: \( \sin^2 x + \cos^2 x = 1 \)

This relationship holds true for any angle \( x \) and is crucial in simplifying trigonometric expressions.
For instance, in the solution to the exercise, the identity was used to express \( \sin^2 x - 1 \) as \( -\cos^2 x \).
This step was essential as it allowed the fraction \( \frac{\sin^2 x - 1}{\cos^2 x} \) to simplify into \( -1 \).
By fully understanding the Pythagorean identity, solving trigonometric problems becomes more manageable, as it provides a path to simplify expressions quickly and accurately.