Find the value of secθ - tanθ, if secθ + tanθ = \(\sqrt5\).
Finding the Value of secθ - tanθ
This problem requires us to find the value of the expression \( \sec\theta - \tan\theta \) given that \( \sec\theta + \tan\theta = \sqrt{5} \). We will use a key trigonometric identity to solve this efficiently.
Understanding the Relationship Between Secant and Tangent
A fundamental trigonometric identity connects secant and tangent:
\[ \sec^2\theta - \tan^2\theta = 1 \]This identity is always true for angles \( \theta \) where \( \sec\theta \) and \( \tan\theta \) are defined.
Applying the Difference of Squares Identity
The left side of the identity \( \sec^2\theta - \tan^2\theta \) is in the form of a difference of squares, \( a^2 - b^2 \), which can be factored as \( (a - b)(a + b) \). Applying this to our identity:
\[ \sec^2\theta - \tan^2\theta = (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) \]So, the trigonometric identity becomes:
\[ (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 \]Using the Given Information to Solve
We are given that \( \sec\theta + \tan\theta = \sqrt{5} \). We can substitute this value into the factored identity:
\[ (\sec\theta - \tan\theta)(\sqrt{5}) = 1 \]Now, to find the value of \( \sec\theta - \tan\theta \), we can divide both sides of the equation by \( \sqrt{5} \):
\[ \sec\theta - \tan\theta = \frac{1}{\sqrt{5}} \]Rationalizing the Result
To present the answer in a standard form, we should rationalize the denominator. We multiply the numerator and the denominator by \( \sqrt{5} \):
\[ \sec\theta - \tan\theta = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \] \[ \sec\theta - \tan\theta = \frac{\sqrt{5}}{5} \]Thus, the value of \( \sec\theta - \tan\theta \) is \( \frac{\sqrt{5}}{5} \).
Key Steps Reviewed
- Start with the given equation: \( \sec\theta + \tan\theta = \sqrt{5} \).
- Recall the identity: \( \sec^2\theta - \tan^2\theta = 1 \).
- Factor the identity: \( (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 \).
- Substitute the given value into the factored identity.
- Solve the resulting equation for \( \sec\theta - \tan\theta \).
- Rationalize the denominator to get the final simplified answer.
| Given | Identity Used | Result |
|---|---|---|
| \( \sec\theta + \tan\theta = \sqrt{5} \) | \( \sec^2\theta - \tan^2\theta = 1 \) | \( \sec\theta - \tan\theta = \frac{\sqrt{5}}{5} \) |
Revision Table: Important Trigonometric Identities
| Identity Type | Identity | Relation |
|---|---|---|
| Pythagorean | \( \sin^2\theta + \cos^2\theta = 1 \) | Basic identity |
| Derived Pythagorean | \( \sec^2\theta - \tan^2\theta = 1 \) | Derived by dividing \( \sin^2\theta + \cos^2\theta = 1 \) by \( \cos^2\theta \) |
| Derived Pythagorean | \( \csc^2\theta - \cot^2\theta = 1 \) | Derived by dividing \( \sin^2\theta + \cos^2\theta = 1 \) by \( \sin^2\theta \) |
| Reciprocal | \( \sec\theta = \frac{1}{\cos\theta} \) | Reciprocal of cosine |
| Ratio | \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) | Ratio of sine to cosine |
Additional Information: Reciprocal Relationship
A useful observation from this problem is the reciprocal relationship between \( \sec\theta + \tan\theta \) and \( \sec\theta - \tan\theta \). From the identity \( (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 \), it directly follows that:
\[ \sec\theta - \tan\theta = \frac{1}{\sec\theta + \tan\theta} \]and also
\[ \sec\theta + \tan\theta = \frac{1}{\sec\theta - \tan\theta} \]This means if you are given the value of one expression (like \( \sec\theta + \tan\theta = \sqrt{5} \)), the value of the other expression \( \sec\theta - \tan\theta \) is simply its reciprocal (\( \frac{1}{\sqrt{5}} \), which rationalizes to \( \frac{\sqrt{5}}{5} \)). This reciprocal relationship is a direct consequence of the \( \sec^2\theta - \tan^2\theta = 1 \) identity and is often used in solving problems involving these terms.
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