Understanding the Problem

The problem asks us to find an expression for \(\tan^2 \theta\) given an equation that involves \(\sin^2 \theta\) and \(\cos^2 \theta\). The given equation is \(a \sin^2 \theta + b \cos^2 \theta = c\).

Our goal is to manipulate this equation using trigonometric identities to isolate and determine the value of \(\tan^2 \theta\) in terms of the constants \(a\), \(b\), and \(c\).

Key Trigonometric Identities

To successfully solve this problem, we will rely on the fundamental relationships between trigonometric functions. The crucial identities required are:

• Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\). This identity helps us relate sine and cosine.
• Tangent Definition: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Squaring this gives us \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\).
• Secant Definition: \(\sec \theta = \frac{1}{\cos \theta}\). Squaring this gives us \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\).
• Secant-Tangent Identity: \(\sec^2 \theta = 1 + \tan^2 \theta\). This identity directly links \(\sec^2 \theta\) with \(\tan^2 \theta\).

Step-by-Step Derivation of tan² θ

Let's begin with the given equation:

\[a \sin^2 \theta + b \cos^2 \theta = c\]

To introduce \(\tan^2 \theta\), which is \(\frac{\sin^2 \theta}{\cos^2 \theta}\), a very effective strategy is to divide every term in the entire equation by \(\cos^2 \theta\). This operation is valid as long as \(\cos^2 \theta \neq 0\).

Dividing each term by \(\cos^2 \theta\):

\[\frac{a \sin^2 \theta}{\cos^2 \theta} + \frac{b \cos^2 \theta}{\cos^2 \theta} = \frac{c}{\cos^2 \theta}\]

Now, we can simplify each term using our trigonometric identities:

• The first term, \(\frac{\sin^2 \theta}{\cos^2 \theta}\), simplifies to \(\tan^2 \theta\).
• The second term, \(\frac{\cos^2 \theta}{\cos^2 \theta}\), simplifies to \(1\).
• The third term, \(\frac{1}{\cos^2 \theta}\), simplifies to \(\sec^2 \theta\).

Substituting these simplified forms back into our equation, we get:

\[a \tan^2 \theta + b(1) = c \sec^2 \theta\]

\[a \tan^2 \theta + b = c \sec^2 \theta\]

At this point, we have \(\tan^2 \theta\) and \(\sec^2 \theta\) in the equation. Our goal is to express everything in terms of \(\tan^2 \theta\). We use the identity \(\sec^2 \theta = 1 + \tan^2 \theta\) to replace \(\sec^2 \theta\):

\[a \tan^2 \theta + b = c (1 + \tan^2 \theta)\]

Next, distribute \(c\) on the right-hand side of the equation:

\[a \tan^2 \theta + b = c + c \tan^2 \theta\]

Now, we need to gather all terms containing \(\tan^2 \theta\) on one side of the equation and all constant terms on the other side. Let's move the \(\tan^2 \theta\) terms to the left side and constant terms to the right side:

Subtract \(c \tan^2 \theta\) from both sides:

\[a \tan^2 \theta - c \tan^2 \theta + b = c\]

Subtract \(b\) from both sides:

\[a \tan^2 \theta - c \tan^2 \theta = c - b\]

Factor out \(\tan^2 \theta\) from the terms on the left side:

\[(a - c) \tan^2 \theta = c - b\]

Finally, to solve for \(\tan^2 \theta\), divide both sides of the equation by \((a - c)\), assuming \(a - c \neq 0\):

\[\tan^2 \theta = \frac{c - b}{a - c}\]

This is the required expression for \(\tan^2 \theta\).

Comparison with Options

Let's check which of the provided options matches our derived expression for \(\tan^2 \theta\):

The derived result \(\frac{c-b}{a-c}\) matches Option 2 perfectly.

Final Answer

The value of \(\tan^2 \theta\) is \(\frac{c-b}{a-c}\).