We are given the equation:

$\frac{1}{\sec \theta} + \tan \theta = \sec \theta - \tan \theta$

Let's try to solve it step-by-step:

Step 1: Simplify $\frac{1}{\sec \theta}$

We know that $\sec \theta = \frac{1}{\cos \theta}$, so $\frac{1}{\sec \theta} = \cos \theta$.

Now the equation becomes:

$\cos \theta + \tan \theta = \sec \theta - \tan \theta$

Step 2: Express $\tan \theta$

We can express $\tan \theta$ in terms of sine and cosine:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

Substituting this into the equation:

$\cos \theta + \frac{\sin \theta}{\cos \theta} = \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}$

Step 3: Combine terms

Both sides of the equation have a denominator of $\cos \theta$, so we can simplify:

Left-hand side (LHS): $\frac{\cos^2 \theta + \sin \theta}{\cos \theta}$

Right-hand side (RHS): $\frac{1 - \sin \theta}{\cos \theta}$

So the equation becomes:

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

Step 4: Multiply both sides by $\cos \theta$

This eliminates the denominators:

$\cos^2 \theta + \sin \theta = 1 - \sin \theta$

Step 5: Move terms around

Move all terms to one side of the equation:

$\cos^2 \theta + \sin \theta + \sin \theta - 1 = 0$

This simplifies to:

$\cos^2 \theta + 2\sin \theta - 1 = 0$

Step 6: Use the identity $\cos^2 \theta = 1 - \sin^2 \theta$

Substitute $\cos^2 \theta$ with $1 - \sin^2 \theta$:

$1 - \sin^2 \theta + 2\sin \theta - 1 = 0$

This simplifies to:

$-\sin^2 \theta + 2\sin \theta = 0$

Step 7: Factor the equation

$-\sin \theta (\sin \theta - 2) = 0$

This gives two possible solutions:

1. $\sin \theta = 0$
2. $\sin \theta = 2$ (which is not possible since the sine function has values between -1 and 1)

Step 8: Solve for $\theta$

For $\sin \theta = 0$, $\theta = n\pi$, where $n$ is an integer.

Final Solution:

$\theta = n\pi, \quad n \in \mathbb{Z}$

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the properties of the sine and cosine functions?
2. How can we use trigonometric identities to simplify expressions?
3. What are the general solutions for $\sin \theta = 0$?
4. How does the secant function relate to cosine?
5. Why is it important to express tangent in terms of sine and cosine?

Tip: Always check the domain of trigonometric functions to avoid invalid solutions.