Problem 48 - Algebra 2

Question

Verify that each of the following is an identity. \(\sin ^{2} \theta+\tan ^{2} \theta=\left(1-\cos ^{2} \theta\right)+\frac{\sec ^{2} \theta}{\csc ^{2} \theta}\)

Step-by-Step Solution

Answer

The expression is an identity as both sides are equal.

Step 1: Rewrite Known Identities

First, let's rewrite the known trigonometric identities. We know that \( \sin^2 \theta = 1 - \cos^2 \theta \) and \( \sec^2 \theta = 1 + \tan^2 \theta \). We also know \( \csc \theta = \frac{1}{\sin \theta} \), so \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \).

Step 2: Simplify Right Side

Substitute the identity \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \) into the expression \( \frac{\sec^2 \theta}{\csc^2 \theta} \). Therefore, \( \frac{\sec^2 \theta}{\csc^2 \theta} = \sec^2 \theta \cdot \sin^2 \theta = (1 + \tan^2 \theta)\sin^2 \theta \).

Step 3: Combine and Simplify

By substituting \( \sec^2 \theta = 1 + \tan^2 \theta \), the term \( \frac{\sec^2 \theta}{\csc^2 \theta} \) becomes \( (1 + \tan^2 \theta) \sin^2 \theta = \sin^2 \theta + \sin^2 \theta \tan^2 \theta \).

Step 4: Compare Both Sides

Now let's compare both sides:- Left side: \( \sin^2 \theta + \tan^2 \theta \).- Right side: \( (1 - \cos^2 \theta) + \sin^2 \theta + \tan^2 \theta \sin^2 \theta \).Based on the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we combine terms on the right:\( 1 - \cos^2 \theta + \sin^2 \theta = \sin^2 \theta + \tan^2 \theta \).Thus, both sides are equal.

Key Concepts

Sine and Cosine

The sine and cosine functions are fundamental in trigonometry, representing the coordinates of a point on the unit circle. The relationship \(\sin^2 \theta + \cos^2 \theta = 1\) is known as the Pythagorean identity. This identity illustrates that the square of the sine function added to the square of the cosine function equals 1.
This identity is crucial because it helps relate the two functions directly and it serves as the foundation for various other trigonometric identities.

• Sine (\(\sin \theta\)): Represents the y-coordinate of a point on the unit circle and is opposite the angle \(\theta\).
• Cosine (\(\cos \theta\)): Represents the x-coordinate of a point on the unit circle and is adjacent to the angle \(\theta\).

Knowing these relationships allows us to manipulate and verify complex trigonometric expressions. For example, rewriting \(\sin^2 \theta = 1 - \cos^2 \theta\) becomes useful in simplifying and verifying identities.

Tangent and Secant

Tangent and secant are two other important trigonometric functions. The tangent of an angle \(\theta\) is defined as the ratio of the sine to the cosine, while the secant is the reciprocal of the cosine.

• Tangent (\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)) : This function helps to relate angles and side ratios in a right triangle.
• Secant (\(\sec \theta = \frac{1}{\cos \theta}\)) : It is the reciprocal of the cosine function and highlights angles where cosine approaches zero.

In verifying identities, the identity \(\sec^2 \theta = 1 + \tan^2 \theta\) is particularly important. It provides a direct link between these functions, enabling simplification of trigonometric equations. Understanding these relationships allows us to substitute or transform parts of an equation into equivalent forms, thus helping to confirm identities.

Cosecant

Cosecant is often less familiar than sine or cosine but is equally important in trigonometry. It is the reciprocal of the sine function.

• Cosecant (\(\csc \theta = \frac{1}{\sin \theta}\)) : Used when dealing with scenarios where the sine function approaches zero.

When verifying a trigonometric identity, transformations involving cosecant can simplify complex expressions. For instance, writing \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\) helps in dividing sequences of functions in equations. These transformations allow us to simplify expressions or compute results more directly.
In our verified identity, converting \(\frac{\sec^2 \theta}{\csc^2 \theta}\) to \(\sec^2 \theta \cdot \sin^2 \theta\) shows how these transformations help in reconciling both sides of an identity.