To solve the equation \(\frac{\tan x}{\sec x - 1} = \frac{\sec x + 1}{\tan x}\), we will first express everything in terms of sine and cosine, then simplify both sides and check for equality.

First, recall the trigonometric identities:
\[
\tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \sec x = \frac{1}{\cos x}
\]

Using these identities, rewrite the given equation:

### Left-hand side (LHS):
\[
\frac{\tan x}{\sec x - 1} = \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x} - 1}
\]

Simplify the denominator:
\[
\frac{1}{\cos x} - 1 = \frac{1 - \cos x}{\cos x}
\]

So, the LHS becomes:
\[
\frac{\frac{\sin x}{\cos x}}{\frac{1 - \cos x}{\cos x}} = \frac{\sin x}{1 - \cos x}
\]

### Right-hand side (RHS):
\[
\frac{\sec x + 1}{\tan x} = \frac{\frac{1}{\cos x} + 1}{\frac{\sin x}{\cos x}}
\]

Simplify the numerator:
\[
\frac{1 + \cos x}{\cos x}
\]

So, the RHS becomes:
\[
\frac{\frac{1 + \cos x}{\cos x}}{\frac{\sin x}{\cos x}} = \frac{1 + \cos x}{\sin x}
\]

### Equating LHS and RHS:

We now have:
\[
\frac{\sin x}{1 - \cos x} = \frac{1 + \cos x}{\sin x}
\]

Cross-multiply to solve for \(x\):
\[
(\sin x)^2 = (1 - \cos x)(1 + \cos x)
\]

Using the identity \(1 - \cos^2 x = \sin^2 x\), we have:
\[
\sin^2 x = 1 - \cos^2 x
\]

Both sides are equal:
\[
\sin^2 x = \sin^2 x
\]

Therefore, the equation is true for all values of \(x\) where the original expressions are defined. 

Would you like details or have any questions? Here are some possible next questions:

1. Can we find specific values of \(x\) that satisfy the equation?
2. What are the steps to simplify trigonometric expressions?
3. How do we verify trigonometric identities step-by-step?
4. What other trigonometric identities are useful for solving equations?
5. Can we explore another example of a trigonometric equation?

**Tip:** Always remember to express trigonometric functions in terms of sine and cosine to simplify and solve equations effectively.

Learn how to solve the trigonometric equation tan x / sec x -1 = (sec x + 1)/ tan x step-by-step with detailed explanations and applications of trigonometric identities. Suitable for students in Grades 10-12.

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