Solving Trigonometric Expressions with Secant, Tangent, and Sine

The problem asks us to find the value of a given trigonometric expression involving \(\tan^2\theta\) and \(\sin^2\theta\), given that \(sec\theta = \frac{a}{b}\). To solve this, we need to express \(\tan^2\theta\) and \(\sin^2\theta\) in terms of \(a\) and \(b\) using the given information and fundamental trigonometric identities.

Step-by-Step Solution Process

We are given:

• \(sec\theta = \frac{a}{b}\), where \(b \ne 0\).

We need to evaluate the expression:

\(\frac{{1 - {{\tan }^2}\theta }}{{2 - {{\sin }^2}\theta }}\)

1. Find \(cos\theta\) from \(sec\theta\)

The secant function is the reciprocal of the cosine function. Therefore:

\(cos\theta = \frac{1}{sec\theta} = \frac{1}{\frac{a}{b}} = \frac{b}{a}\)

2. Find \({\sin }^2\theta\) using Trigonometric Identity

We use the fundamental Pythagorean identity: \({\sin }^2\theta + {\cos }^2\theta = 1\). We can rearrange this to solve for \({\sin }^2\theta\):

\({\sin }^2\theta = 1 - {\cos }^2\theta\)

Substitute the value of \(cos\theta\):

\({\sin }^2\theta = 1 - {\left(\frac{b}{a}\right)}^2 = 1 - \frac{b^2}{a^2}\)

Combine the terms on the right side:

\({\sin }^2\theta = \frac{a^2}{a^2} - \frac{b^2}{a^2} = \frac{a^2 - b^2}{a^2}\)

3. Find \({\tan }^2\theta\) using Trigonometric Identity

We use another Pythagorean identity: \(1 + {\tan }^2\theta = {\sec }^2\theta\). We can rearrange this to solve for \({\tan }^2\theta\):

\({\tan }^2\theta = {\sec }^2\theta - 1\)

Substitute the given value of \(sec\theta\):

\({\tan }^2\theta = {\left(\frac{a}{b}\right)}^2 - 1 = \frac{a^2}{b^2} - 1\)

Combine the terms on the right side:

\({\tan }^2\theta = \frac{a^2}{b^2} - \frac{b^2}{b^2} = \frac{a^2 - b^2}{b^2}\)

4. Substitute values into the Expression

Now we substitute the expressions for \({\tan }^2\theta\) and \({\sin }2\theta\) back into the original expression \(\frac{{1 - {{\tan }^2}\theta }}{{2 - {{\sin }^2}\theta }}\).

Numerator: \(1 - {\tan }^2\theta\)

\(1 - {\tan }^2\theta = 1 - \left(\frac{a^2 - b^2}{b^2}\right)\)

\( = \frac{b^2}{b^2} - \frac{a^2 - b^2}{b^2} = \frac{b^2 - (a^2 - b^2)}{b^2} = \frac{b^2 - a^2 + b^2}{b^2} = \frac{2b^2 - a^2}{b^2}\)

Denominator: \(2 - {\sin }^2\theta\)

\(2 - {\sin }^2\theta = 2 - \left(\frac{a^2 - b^2}{a^2}\right)\)

\( = \frac{2a^2}{a^2} - \frac{a^2 - b^2}{a^2} = \frac{2a^2 - (a^2 - b^2)}{a^2} = \frac{2a^2 - a^2 + b^2}{a^2} = \frac{a^2 + b^2}{a^2}\)

5. Simplify the Complex Fraction

Now we have the expression as a fraction divided by another fraction:

\(\frac{{\frac{2b^2 - a^2}{b^2}}}{{\frac{a^2 + b^2}{a^2}}}\)

To divide by a fraction, we multiply by its reciprocal:

\(= \frac{2b^2 - a^2}{b^2} \times \frac{a^2}{a^2 + b^2}\)

Multiply the numerators and the denominators:

\(= \frac{(2b^2 - a^2) \times a^2}{b^2 \times (a^2 + b^2)}\)

\(= \frac{a^2(2b^2 - a^2)}{b^2(a^2 + b^2)}\)

This simplified expression matches one of the given options.

Comparing with Options

Let's look at the options provided:

• Option 1: \(\frac{{{b^2}\left( {2{b^2} - {a^2}} \right)}}{{{a^2}\left( {{a^2} + {b^2}} \right)}}\)
• Option 2: \(\frac{{{b^2}\left( {2{b^2} + {a^2}} \right)}}{{{a^2}\left( {{a^2} + {b^2}} \right)}}\)
• Option 3: \(\frac{{{a^2}\left( {2{b^2} - {a^2}} \right)}}{{{b^2}\left( {{a^2} + {b^2}} \right)}}\)
• Option 4: \(\frac{{{a^2}\left( {2{b^2} - {a^2}} \right)}}{{{a^2}\left( {{a^2} - {b^2}} \right)}}\)

Our derived expression is \(\frac{a^2(2b^2 - a^2)}{b^2(a^2 + b^2)}\), which exactly matches Option 3.

Revision Table: Key Concepts Revisited

Additional Information on Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables where both sides of the equation are defined. They are essential tools for simplifying trigonometric expressions and solving trigonometric equations.

• Reciprocal Identities: These relate the six trigonometric functions in pairs.
  • \(sin\theta = \frac{1}{csc\theta}\)
  • \(cos\theta = \frac{1}{sec\theta}\)
  • \(tan\theta = \frac{1}{cot\theta}\)
• Quotient Identities: These express tangent and cotangent in terms of sine and cosine.
  • \(tan\theta = \frac{sin\theta}{cos\theta}\)
  • \(cot\theta = \frac{cos\theta}{sin\theta}\)
• Pythagorean Identities: These are derived from the Pythagorean theorem and are fundamental.
  • \({\sin }^2\theta + {\cos }^2\theta = 1\)
  • \(1 + {\tan }^2\theta = {\sec }^2\theta\)
  • \(1 + {\cot }^2\theta = {\csc }^2\theta\)

In this problem, we effectively used reciprocal identities and two of the Pythagorean identities to convert the given expression into terms of \(a\) and \(b\) from the initial condition \(sec\theta = \frac{a}{b}\).