Problem 80 - Algebra and Trigonometry

Question

Verify the identity. $$ \tan ^{2} x-\cot ^{2} x=\sec ^{2} x-\csc ^{2} x $$

Step-by-Step Solution

Answer

The identity is verified by utilizing Pythagorean identities and simplifying both sides.

Step 1: Use Pythagorean Identities

Recall the Pythagorean identities: \( \tan^2 x + 1 = \sec^2 x \) and \( \cot^2 x + 1 = \csc^2 x \). We'll use these identities to rewrite both sides of the given equation.

Step 2: Transform \( \tan^2 x - \cot^2 x \)

Using the identities from Step 1, we have \( \tan^2 x = \sec^2 x - 1 \) and \( \cot^2 x = \csc^2 x - 1 \). Substitute these into \( \tan^2 x - \cot^2 x \) to get: \[ \tan^2 x - \cot^2 x = (\sec^2 x - 1) - (\csc^2 x - 1) \]. Simplify this to get \( \tan^2 x - \cot^2 x = \sec^2 x - \csc^2 x \).

Step 3: Verify by Simplification

Simplifying the expression \( (\sec^2 x - 1) - (\csc^2 x - 1) \) yields \( \sec^2 x - \csc^2 x \), which matches the right-hand side of the identity \( \sec^2 x - \csc^2 x \). This confirms that the identity \( \tan^2 x - \cot^2 x = \sec^2 x - \csc^2 x \) holds true.

Key Concepts

Pythagorean Identities

The Pythagorean identities are foundational in trigonometry, similar to how the Pythagorean theorem is foundational in geometry. These identities relate the trigonometric functions to one another and are derived from the basic relationship of a right triangle's sides. They help simplify complex trigonometric expressions and verify identities.
The key Pythagorean identities are:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( \tan^2 x + 1 = \sec^2 x \)
• \( \cot^2 x + 1 = \csc^2 x \)

These identities allow us to express sine, cosine, tangent, cotangent, secant, and cosecant in terms of each other. For solving problems, such as our exercise here, they are indispensable tools. Knowing them by heart can significantly aid in both simplifying and transforming trigonometric expressions.

Tangent and Cotangent

The tangent and cotangent functions are reciprocal relations and have unique identities tied to the circle and angles. Recognizing how these functions relate to each other is crucial in solving identities and equations in trigonometry.

• Tangent function: \( \tan x = \frac{\sin x}{\cos x} \)
• Cotangent function: \( \cot x = \frac{\cos x}{\sin x} \)
• Identity: \( \tan^2 x + 1 = \sec^2 x \)
• Identity: \( \cot^2 x + 1 = \csc^2 x \)

Using these relationships, we can substitute values in identities, like in our given exercise. Here, we understand that the expression \( \tan^2 x - \cot^2 x \) simplifies because of their respective identities with secant and cosecant which we already used. This systematic method of applying identities allows us to verify trigonometric equations efficiently.

Secant and Cosecant

Secant and cosecant are the reciprocal functions of cosine and sine, respectively. These functions might seem less straightforward, so it's helpful to understand how they work with other trigonometric identities.

• Secant: \( \sec x = \frac{1}{\cos x} \)
• Cosecant: \( \csc x = \frac{1}{\sin x} \)
• Identity for secant: \( \sec^2 x = \tan^2 x + 1 \)
• Identity for cosecant: \( \csc^2 x = \cot^2 x + 1 \)

In our exercise, we used these identities to simplify and transform the equation from one form to another, ultimately verifying the correctness of the trigonometric identity \( \tan^2 x - \cot^2 x = \sec^2 x - \csc^2 x \). By expressing tangent and cotangent in terms of secant and cosecant, we simplified both sides of the equation, making them equal, thus proving the identity true.