FOR ANY ACUTE ANGLE R 1 +(cot^2R/1+cosecR

The expression you provided is:

\[
1 + \left(\frac{\cot^2 R}{1 + \csc R}\right)
\]

Let’s break it down and simplify the expression step by step.

### Step 1: Use trigonometric identities
Recall the following trigonometric identities:
1. \(\cot R = \frac{\cos R}{\sin R}\)
2. \(\csc R = \frac{1}{\sin R}\)

Now, substitute these identities into the expression:

\[
1 + \left(\frac{\left(\frac{\cos^2 R}{\sin^2 R}\right)}{1 + \frac{1}{\sin R}}\right)
\]

### Step 2: Simplify the denominator
The denominator inside the fraction is \(1 + \frac{1}{\sin R}\). We can combine this into a single fraction:

\[
1 + \frac{1}{\sin R} = \frac{\sin R + 1}{\sin R}
\]

So now the expression becomes:

\[
1 + \left(\frac{\frac{\cos^2 R}{\sin^2 R}}{\frac{\sin R + 1}{\sin R}}\right)
\]

### Step 3: Simplify the entire fraction
To simplify the complex fraction, multiply by the reciprocal of the denominator:

\[
\frac{\frac{\cos^2 R}{\sin^2 R}}{\frac{\sin R + 1}{\sin R}} = \frac{\cos^2 R}{\sin^2 R} \times \frac{\sin R}{\sin R + 1}
\]

This simplifies to:

\[
\frac{\cos^2 R}{\sin R (\sin R + 1)}
\]

### Step 4: Final expression
Now, the original expression becomes:

\[
1 + \frac{\cos^2 R}{\sin R (\sin R + 1)}
\]

This is the simplified form of the given trigonometric expression.

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Do you have any questions about the process or want further details?

Here are 5 related questions for further exploration:
1. How can you use Pythagorean identities to simplify trigonometric expressions?
2. What are some other forms of expressing \(\cot R\) and \(\csc R\)?
3. How can you express \(\sin^2 R + \cos^2 R\) in terms of \(\tan R\)?
4. How does this expression behave as \(R \to 0\)?
5. Can you derive a similar expression using \(\tan R\) instead of \(\cot R\)?

**Tip:** Trigonometric identities often involve fractions; simplifying by common denominators and using known identities makes the process easier.

FOR ANY ACUTE ANGLE R 1 +(cot^2R/1+cosecR)

This page provides a detailed step-by-step solution to simplify the trigonometric expression 1 + (cot^2 R / 1 + csc R), using key trigonometric identities. Learn about cotangent, cosecant, and Pythagorean identities, suitable for high school students in Grades 10-12.

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