Problem 4 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{1+\tan ^{2} \theta}{\csc ^{2} \theta}=\tan ^{2} \theta $$

Step-by-Step Solution

Answer

The identity is valid; both sides simplify to \( \frac{\sin^2 \theta}{\cos^2 \theta} \).

Step 1: Understand the Identity to be Verified

We need to verify the trigonometric identity \( \frac{1+\tan^2 \theta}{\csc^2 \theta} = \tan^2 \theta \). This means we must show that both sides of the equation are equivalent for all valid \( \theta \).

Step 2: Recall Trigonometric Identities

Let's recall some important trigonometric identities that will be useful:- \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \)- \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \)- The Pythagorean identity: \( 1 + \tan^2 \theta = \sec^2 \theta \) where \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \).

Step 3: Simplify the Left Side of the Equation

Substitute \( 1 + \tan^2 \theta = \sec^2 \theta \) into the left side:\[ \frac{\sec^2 \theta}{\csc^2 \theta} = \frac{\frac{1}{\cos^2 \theta}}{\frac{1}{\sin^2 \theta}} \]This simplifies to:\[ \frac{\sec^2 \theta}{\csc^2 \theta} = \frac{\sin^2 \theta}{\cos^2 \theta} \]

Step 4: Identify the Right Side of the Equation

The right side of the equation is already \( \tan^2 \theta \), which can be expressed as \( \frac{\sin^2 \theta}{\cos^2 \theta} \), identical to the simplified left side.

Step 5: Conclude the Identity

Both sides of the original equation simplify to \( \frac{\sin^2 \theta}{\cos^2 \theta} \), thereby confirming the identity. Since both sides are equal, the identity \( \frac{1+\tan^2 \theta}{\csc^2 \theta} = \tan^2 \theta \) is verified.

Key Concepts

Pythagorean identity

The Pythagorean identity is a central concept in trigonometry, similar in importance to the Pythagorean theorem in geometry. It establishes a relationship between the trigonometric functions sine, cosine, and tangent. The most commonly used form of the Pythagorean identity is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity can be manipulated to derive additional identities such as \(1 + \tan^2 \theta = \sec^2 \theta\).
The beauty of the Pythagorean identity lies in its ability to simplify and verify various trigonometric expressions through these conversions. Given its fundamental nature, it appears frequently in different forms while solving trigonometric equations and verifying identities. It is vital for students to remember this identity and its transformations. Without the Pythagorean identity, simplifying expressions like those in this exercise would be much more challenging.

Secant

The secant function, written as \( \sec \theta \), is one of the reciprocal trigonometric functions. It is the reciprocal of the cosine function, meaning:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Understanding the secant function is crucial when working with expressions involving these reciprocal relationships.
It plays a significant role, especially when expressions involve \( \sec^2 \theta \), like in the provided identity verification.
In the given problem, recognizing the transformation \( 1 + \tan^2 \theta = \sec^2 \theta \) allows us to simplify the left side of the equation effectively. This highlights the importance of knowing the role of secant in verifying trigonometric identities.

Cosecant

Cosecant, represented as \( \csc \theta \), is another reciprocal trigonometric function. It is the reciprocal of the sine function, defined as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

In solving trigonometric identities, knowing that \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \) helps in converting and simplifying expressions, such as in the exercise we are analyzing.
When working through the step-by-step solution, the transformation of \( \csc^2 \theta \) to \( \frac{1}{\sin^2 \theta} \) simplifies the original expression and aids in reaching the solution. It's beneficial to understand how the cosecant function assists in breaking down complex trigonometric equations into simpler components.

Tangent

The tangent function, represented as \( \tan \theta \), is one of the primary trigonometric functions most commonly encountered in trigonometry. It can be expressed in terms of sine and cosine:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This relationship is key in understanding the trigonometric identities and simplifications needed in exercises like the one provided.
In the given exercise, \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \) is used to equate the expression to its simplified form. Mastery of how tangent relates to other trigonometric functions enables smoother transitions during simplification and verification processes.

Simplification of trigonometric expressions

Simplification of trigonometric expressions is a critical skill when dealing with trigonometric identities and equations. It involves the application of various trigonometric identities and relationships to transform a complex expression into a simpler, more manageable form.

• Recognizing how different trigonometric functions relate to one another is crucial.
• Using known identities, like the Pythagorean identity \(1 + \tan^2 \theta = \sec^2 \theta\), helps streamline lengthy or complicated expressions.
• Converting reciprocal functions like \(\csc^2 \theta\) to \(\frac{1}{\sin^2 \theta}\) and \(\sec^2 \theta\) to \(\frac{1}{\cos^2 \theta}\) can drastically simplify the problem.

The solution provided in the exercise is a perfect illustration of this process. By substituting known identities and reducing fractions, the originally complex expression is shown to be equal on both sides, ultimately proving the identity.