(1 + sin x) / 1 - sin x = 2 sec^ 2 x + 2 sec x * tan x -1

To solve the equation \(\frac{1 + \sin x}{1 - \sin x} = 2 \sec^2 x + 2 \sec x \tan x - 1\), let's go through the steps to verify and simplify both sides of the equation. 

First, we work on the left-hand side (LHS):

\[ \text{LHS} = \frac{1 + \sin x}{1 - \sin x} \]

We can use the identity \(\frac{1 + \sin x}{1 - \sin x} = \left( \frac{1 + \sin x}{1 - \sin x} \right) \cdot \left( \frac{1 + \sin x}{1 + \sin x} \right) = \frac{(1 + \sin x)^2}{(1 - \sin x)(1 + \sin x)} \).

\[ \text{LHS} = \frac{1 + 2 \sin x + \sin^2 x}{1 - \sin^2 x} \]

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

\[ \text{LHS} = \frac{1 + 2 \sin x + \sin^2 x}{\cos^2 x} \]
\[ \text{LHS} = \frac{1 + 2 \sin x + \sin^2 x}{\cos^2 x} = \sec^2 x + 2 \sec x \tan x \]

Now let's simplify the right-hand side (RHS):

\[ \text{RHS} = 2 \sec^2 x + 2 \sec x \tan x - 1 \]

To check if the two expressions are equal, let's rewrite them and compare:

\[ \sec^2 x + 2 \sec x \tan x \]
\[ 2 \sec^2 x + 2 \sec x \tan x - 1 \]

Clearly, there seems to be a discrepancy between the simplified left-hand side and the right-hand side. Since \(\sec^2 x + 2 \sec x \tan x\) does not simplify directly to \(2 \sec^2 x + 2 \sec x \tan x - 1\).

Therefore, there seems to be a mistake in the original equation. The equation \(\frac{1 + \sin x}{1 - \sin x} = 2 \sec^2 x + 2 \sec x \tan x - 1\) does not hold true as the LHS simplifies to \(\sec^2 x + 2 \sec x \tan x\) while the RHS is \(2 \sec^2 x + 2 \sec x \tan x - 1\).

Do you have any specific steps or another method you'd like to explore for solving or verifying this equation?

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

1. Can we solve \(\sec^2 x + 2 \sec x \tan x = 2 \sec^2 x + 2 \sec x \tan x - 1\) for specific values of \(x\)?
2. What are the identities involving \(\sec x\) and \(\tan x\)?
3. Can we explore another approach to simplify \(\frac{1 + \sin x}{1 - \sin x}\)?
4. How do we derive the identity \(\sec^2 x - \tan^2 x = 1\)?
5. What are the applications of \(\sec x\) and \(\tan x\) in trigonometry?

**Tip:** Always verify trigonometric identities step-by-step to ensure accuracy in simplification and solving equations.

Explore how to solve the trigonometric equation (1 + sin x) / (1 - sin x) = 2 sec^2 x + 2 sec x tan x - 1. This problem delves into advanced trigonometric identities and equations, with detailed step-by-step solutions and explanations.

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