Understanding the Trigonometric Expression

The question asks us to simplify the given trigonometric expression:

$$ \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} $$

We need to find which of the given options is equal to this expression. Simplifying trigonometric expressions often involves using known identities and algebraic manipulation.

Step-by-Step Simplification Process

To simplify the expression $\frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta}$, a common technique is to multiply the numerator and the denominator by the conjugate of the denominator. The denominator is $(\sec\theta - \tan\theta)$, so its conjugate is $(\sec\theta + \tan\theta)$.

Let's perform the multiplication:

$$ \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} \times \frac{\sec\theta + \tan\theta}{\sec\theta + \tan\theta} $$

Now, we simplify the numerator and the denominator separately.

The numerator is $(\sec\theta + \tan\theta) \times (\sec\theta + \tan\theta)$, which is $(\sec\theta + \tan\theta)^2$.

The denominator is $(\sec\theta - \tan\theta) \times (\sec\theta + \tan\theta)$. This is in the form of $(a-b)(a+b)$, which simplifies to $a^2 - b^2$. Here, $a = \sec\theta$ and $b = \tan\theta$.

So, the denominator becomes $(\sec\theta)^2 - (\tan\theta)^2 = \sec^2\theta - \tan^2\theta$.

Our expression now is:

$$ \frac{(\sec\theta + \tan\theta)^2}{\sec^2\theta - \tan^2\theta} $$

Next, we use a fundamental trigonometric identity. We know that $\sec^2\theta - \tan^2\theta = 1$.

Substituting this identity into the denominator, we get:

$$ \frac{(\sec\theta + \tan\theta)^2}{1} $$

Simplifying this, the expression is equal to $(\sec\theta + \tan\theta)^2$.

Comparing with the Options

We simplified the given expression $\frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta}$ to $(\sec\theta + \tan\theta)^2$. Now let's look at the given options:

• Option 1: \(1/(\sec\theta - \tan\theta)\)
• Option 2: \(1/(\sec\theta + \tan\theta)\)
• Option 3: \((\sec\theta + \tan\theta)^2\)
• Option 4: \((\sec\theta - \tan\theta)^2\)

Comparing our result with the options, we see that our simplified expression matches Option 3.

Key Trigonometric Identity Explained

The crucial step in this simplification was using the identity relating $\sec\theta$ and $\tan\theta$. This identity comes directly from the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$.

If we divide the identity $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$ (assuming $\cos\theta \neq 0$), we get:

$$ \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta} $$

Using the definitions $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\sec\theta = \frac{1}{\cos\theta}$, this becomes:

$$ (\tan\theta)^2 + 1 = (\sec\theta)^2 $$

$$ \tan^2\theta + 1 = \sec^2\theta $$

Rearranging this identity gives us:

$$ \sec^2\theta - \tan^2\theta = 1 $$

This identity is valid for all angles $\theta$ where $\sec\theta$ and $\tan\theta$ are defined (i.e., where $\cos\theta \neq 0$).

Additional Information on Trigonometric Simplification Techniques

When faced with simplifying trigonometric expressions, several techniques can be helpful:

• Convert to Sine and Cosine: Expressing everything in terms of $\sin\theta$ and $\cos\theta$ can often simplify complex fractions.
• Use Pythagorean Identities: Look for opportunities to replace $\sin^2\theta + \cos^2\theta$ with 1, $\sec^2\theta - \tan^2\theta$ with 1, or $\csc^2\theta - \cot^2\theta$ with 1 (or variations thereof).
• Factor Expressions: Look for common factors or recognize algebraic patterns like difference of squares ($a^2 - b^2$) or perfect squares ($(a+b)^2$, $(a-b)^2$).
• Find Common Denominators: When adding or subtracting trigonometric fractions, combine them over a common denominator.
• Multiply by the Conjugate: As demonstrated in this problem, multiplying the numerator and denominator by the conjugate of an expression like $(a \pm b)$ can help rationalize the denominator or simplify the expression.

Understanding these techniques and practicing with various problems is key to mastering trigonometric simplification.