Problem 53 - Algebra and Trigonometry

Question

Verify the identity. $$ \tan ^{2} \theta-\sin ^{2} \theta=\tan ^{2} \theta \sin ^{2} \theta $$

Step-by-Step Solution

Answer

The identity is verified; both sides are equal.

Step 1: Rewrite Left Side

Start with the left side of the identity: \( \tan^2 \theta - \sin^2 \theta \). Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), so \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \). Substitute this in to get: \[ \frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta. \]

Step 2: Common Denominator

Express \( \sin^2 \theta \) with a common denominator of \( \cos^2 \theta \) by rewriting it as \( \frac{\sin^2 \theta \cos^2 \theta}{\cos^2 \theta} \). Now, the expression becomes: \[ \frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta \cos^2 \theta}{\cos^2 \theta}. \]

Step 3: Simplify the Expression

Combine the fractions over a common denominator: \[ \frac{\sin^2 \theta - \sin^2 \theta \cos^2 \theta}{\cos^2 \theta}. \] Factor out \( \sin^2 \theta \) from the numerator: \[ \sin^2 \theta \left( 1 - \cos^2 \theta \right). \]

Step 4: Use Pythagorean Identity

Use the identity \( 1 - \cos^2 \theta = \sin^2 \theta \). Substitute this into the expression to get: \[ \frac{\sin^2 \theta \cdot \sin^2 \theta}{\cos^2 \theta}. \]

Step 5: Simplify to Match Right Side

Recognize that the expression \( \frac{\sin^4 \theta}{\cos^2 \theta} \) is \( \tan^2 \theta \sin^2 \theta \) because \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \). Hence, the left side simplifies to \( \tan^2 \theta \sin^2 \theta \).

Step 6: Conclude Verification

The left side \( \tan^2 \theta - \sin^2 \theta \) is equal to the right side \( \tan^2 \theta \sin^2 \theta \). The identity is verified.

Key Concepts

Understanding the Tangent Function

The tangent function is a fundamental trigonometric function defined as the ratio of the sine function to the cosine function. In mathematical form, it's expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship is key when working with trigonometric identities, as it allows us to interconnect various trigonometric expressions, simplifying equations.
The tangent function has some interesting properties:

• Its value is undefined whenever the cosine function equals zero (e.g., at \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \), and so on).
• It has a period of \( \pi \), meaning it repeats its values every \( \pi \) radians.
• Its graph is an odd function, symmetric about the origin.

Understanding \( \tan \theta \) is essential as it is often involved in simplifications and verifications of trigonometric identities, like in the problem given where we express \( \tan^2 \theta \) in terms of \( \sin^2 \theta \) and \( \cos^2 \theta \).

Exploring the Sine Function

The sine function is one of the primary trigonometric functions and it describes the projection of a rotating point's position, typically on the y-axis of a unit circle. Mathematically, it's written as \( \sin \theta \), which can be understood as the y-coordinate of a point on the unit circle.
The sine function has some characteristics:

• It has a range of [-1, 1], representing its highest and lowest possible values.
• Like most trigonometric functions, the sine function is periodic, with a period of \( 2\pi \).
• The graph of the sine function is a smooth, continuous wave that repeats every \( 2\pi \) radians.

In trigonometric identities such as the exercise provided, the sine function plays a vital role, connecting other functions like cosine and tangent through well-known identities.

Pythagorean Identity and Its Applications

The Pythagorean identity is a cornerstone in trigonometry and states that \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is derived from the geometric properties of a right triangle inscribed in a unit circle. This equation is always true for all angles \( \theta \).
Applications of the Pythagorean identity include:

• It helps in transforming trigonometric expressions by substituting \( 1 - \cos^2 \theta \) with \( \sin^2 \theta \) or vice versa. This is precisely what was utilized in step 4 of the original exercise to simplify the expression.
• It aids in calculating unknown function values when one of the sine or cosine values is known.
• It is often used in derivations and proofs within trigonometric identities.

This identity is essential in providing bridges between sine and cosine, simplifying expressions, and expanding the range of solutions for trigonometric equations.

Trigonometric Simplification Techniques

Simplifying trigonometric expressions is a crucial technique in solving equations and verifying identities. The main goal is to reduce complex trigonometric expressions to simpler or equivalent forms.
Here are some techniques used in simplification:

• Common Denominator Approach: As shown in the exercise, transforming different fractions to a common denominator allows us to combine and simplify expressions.
• Factorization: Factoring out common trigonometric terms is another effective method. This was done in step 3 of the exercise where \( \sin^2 \theta \) was factored out.
• Using Trigonometric Identities: Such as using \( 1 - \cos^2 \theta = \sin^2 \theta \), which can simplify terms significantly.
• Substitution: Often, substituting trigonometric ratios, especially linear ones, helps make a complex problem more manageable.

Mastering these techniques is particularly helpful in recognizing patterns and transforming intricate trigonometric identities into more digestible forms, as illustrated in the step-by-step solution.