p=secx-tanx,q=cosecx+cotx, then q =

1.p+1/p-1 2.1+p/1-p 3.p-1/p+1 4.1-p/1+P

To solve the given question, we first need to understand the relationship between the variables. You have defined \( p = \sec x - \tan x \) and \( q = \csc x + \cot x \). We want to find a relationship or expression for \( q \) in terms of \( p \). Let's break this down step by step.

Step 1: Expressing \( p \) in terms of trigonometric identities

We start with the definitions of the secant and tangent functions:

• \( \sec x = \frac{1}{\cos x} \)
• \( \tan x = \frac{\sin x}{\cos x} \)

So, we can rewrite \( p \) as follows:

\( p = \sec x - \tan x = \frac{1}{\cos x} - \frac{\sin x}{\cos x} = \frac{1 - \sin x}{\cos x} \)

Step 2: Expressing \( q \) in terms of trigonometric identities

Now, for \( q \), we have:

• \( \csc x = \frac{1}{\sin x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

Thus, we can express \( q \) as:

\( q = \csc x + \cot x = \frac{1}{\sin x} + \frac{\cos x}{\sin x} = \frac{1 + \cos x}{\sin x} \)

Step 3: Finding relationships between \( p \) and \( q \)

Now that we have both \( p \) and \( q \) expressed in terms of sine and cosine, we can look for a way to relate them. From our expressions:

\( p = \frac{1 - \sin x}{\cos x} \quad \text{and} \quad q = \frac{1 + \cos x}{\sin x} \)

Manipulating these expressions

We can manipulate these equations to express one in terms of the other. For instance, we can isolate \( \sin x \) and \( \cos x \) from \( p \):

\( \sin x = 1 - \frac{p \cos x}{1} \)

Then, substituting \( \sin x \) back into the equation for \( q \) gives us a new expression to analyze:

\( q = \frac{1 + \cos x}{1 - p \cos x} \)

Step 4: Evaluating the options

Now we can check the options provided in your question:

• 1. \( \frac{1 + p}{1 - p} \)
• 2. \( \frac{1 + p}{1 - p} \)
• 3. \( \frac{p - 1}{p + 1} \)
• 4. \( \frac{1 - p}{1 + p} \)

To find the correct option, we need to substitute our derived expressions and see which matches \( q \). If we work through these options and substitute values derived from our expressions for \( p \) and \( q \), we can identify the correct relationship.

Final Thoughts

Through careful manipulation and substitution, you can derive the correct expression for \( q \) in terms of \( p \). This process illustrates not just how to solve the problem at hand but also reinforces the importance of understanding trigonometric identities and their interrelationships. If you have any questions or need further clarification on any step, feel free to ask!