Trigonometry Problem: Finding sec⁴ θ - tan⁴ θ

This problem asks us to find the value of the expression \(\sec^4 \theta - \tan^4 \theta\), given that \(\sec^2 \theta + \tan^2 \theta = \frac{25}{18}\). We can solve this by using fundamental trigonometric and algebraic identities.

Applying Algebraic and Trigonometric Identities

The expression we need to evaluate is \(\sec^4 \theta - \tan^4 \theta\). This expression is in the form of \(a^2 - b^2\), where \(a = \sec^2 \theta\) and \(b = \tan^2 \theta\).

Using the algebraic identity \(a^2 - b^2 = (a - b)(a + b)\), we can rewrite the expression:

\(\sec^4 \theta - \tan^4 \theta = (\sec^2 \theta)^2 - (\tan^2 \theta)^2\)

Applying the identity \(a^2 - b^2 = (a - b)(a + b)\):

\(\sec^4 \theta - \tan^4 \theta = (\sec^2 \theta - \tan^2 \theta)(\sec^2 \theta + \tan^2 \theta)\)

Now, we need to evaluate each factor in the product:

• The second factor is \(\sec^2 \theta + \tan^2 \theta\). The problem statement gives us the value of this factor directly: \(\sec^2 \theta + \tan^2 \theta = \frac{25}{18}\).
• The first factor is \(\sec^2 \theta - \tan^2 \theta\). This is a fundamental trigonometric identity. The relationship between secant and tangent is given by: \(\sec^2 \theta - \tan^2 \theta = 1\).

Substituting the Values

Now we substitute the known values of these two factors back into the expanded expression:

\(\sec^4 \theta - \tan^4 \theta = (\sec^2 \theta - \tan^2 \theta)(\sec^2 \theta + \tan^2 \theta)\)

Substitute \(\sec^2 \theta - tan^2 \theta = 1\) and \(\sec^2 \theta + \tan^2 \theta = \frac{25}{18}\):

\(\sec^4 \theta - \tan^4 \theta = (1)\left(\frac{25}{18}\right)\)

\(\sec^4 \theta - \tan^4 \theta = \frac{25}{18}\)

Thus, the value of \(\sec^4 \theta - \tan^4 \theta\) is \(\frac{25}{18}\).

Step-by-Step Solution Summary

1. Start with the expression \(\sec^4 \theta - \tan^4 \theta\).
2. Recognize this as a difference of squares, \( (a^2)^2 - (b^2)^2 \) where \( a = \sec \theta \) and \( b = \tan \theta \). More simply, recognize it as \( (a)^2 - (b)^2 \) where \( a = \sec^2 \theta \) and \( b = \tan^2 \theta \).
3. Apply the algebraic identity \( a^2 - b^2 = (a - b)(a + b) \) to get \( (\sec^2 \theta - \tan^2 \theta)(\sec^2 \theta + \tan^2 \theta) \).
4. Use the fundamental trigonometric identity \(\sec^2 \theta - \tan^2 \theta = 1\).
5. Substitute the given value \(\sec^2 \theta + \tan^2 \theta = \frac{25}{18}\).
6. Multiply the values: \( (1) \times \left(\frac{25}{18}\right) = \frac{25}{18} \).

Final Result

The value of \(\sec^4 \theta - \tan^4 \theta\) is \(\frac{25}{18}\).

Revision Table: Trigonometric Identities

It's helpful to remember the basic trigonometric identities involving secant and tangent when solving such problems.

• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
• \(\sin^2 \theta + \cos^2 \theta = 1\) (Pythagorean Identity)
• \(\sec^2 \theta - \tan^2 \theta = 1\) (Derived from Pythagorean Identity by dividing by \(\cos^2 \theta\))

Additional Information: Solving Trigonometry Problems

When tackling trigonometry problems, especially those involving powers of trigonometric functions, consider the following approaches:

• Look for opportunities to apply Pythagorean identities like \(\sin^2 \theta + \cos^2 \theta = 1\), \(\sec^2 \theta - \tan^2 \theta = 1\), or \(\csc^2 \theta - \cot^2 \theta = 1\).
• Recognize common algebraic patterns such as difference of squares (\(a^2 - b^2\)) or perfect squares (\((a+b)^2\), \((a-b)^2\)).
• If unsure, try converting all trigonometric functions to their sine and cosine equivalents.
• Simplify expressions step by step, keeping track of the identities used.
• Always check if the given information can be directly substituted into the simplified expression.

This problem was straightforward because the expression \(\sec^4 \theta - \tan^4 \theta\) simplified nicely into factors whose values were either known identities or given in the question.