sec*θ*−tan*θ*=sec*θ*+tan*θ*1​ solve

We are given the equation:

\[
\frac{\sec\theta - \tan\theta}{\sec\theta + \tan\theta} = 1
\]

To solve this, let's proceed step by step:

### Step 1: Multiply both sides by \((\sec\theta + \tan\theta)\)
This eliminates the denominator on the left-hand side:

\[
\sec\theta - \tan\theta = (\sec\theta + \tan\theta)
\]

### Step 2: Simplify the equation
Now we subtract \(\sec\theta + \tan\theta\) from both sides to get all terms on one side:

\[
(\sec\theta - \sec\theta) - (\tan\theta + \tan\theta) = 0
\]

\[
0 = 2\tan\theta
\]

### Step 3: Solve for \(\theta\)
This simplifies to:

\[
\tan\theta = 0
\]

The solutions for \(\tan\theta = 0\) occur when \(\theta = n\pi\), where \(n\) is an integer.

Thus, the solution is:

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

### Would you like to see more details about any of the steps? Here are some related questions:

1. What is the secant and tangent function in terms of sine and cosine?
2. How do you solve trigonometric equations like \(\tan\theta = 0\)?
3. What are the general solutions for trigonometric equations?
4. Can you show the graphs of the \(\sec\theta\) and \(\tan\theta\) functions?
5. Why does \(\theta = n\pi\) satisfy \(\tan\theta = 0\)?

**Tip:** Always remember that the tangent function \(\tan\theta\) is zero at multiples of \(\pi\).

Discover how to solve the trigonometric equation sec(θ) − tan(θ) = sec(θ) + tan(θ) using step-by-step simplification and the tangent function's properties. Learn the solution for θ in terms of nπ and key trigonometric identities, suitable for high school students in Grades 10-12.

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