Simplify The Expression:${ 5 \left(\frac{1}{1-\sin X} + \frac{1}{1+\sin X}\right) = \frac{2}{\cos^2 X} }$

Introduction

Trigonometric identities are a fundamental concept in mathematics, and they play a crucial role in solving various mathematical problems. In this article, we will focus on simplifying a given expression involving trigonometric functions. The expression is:

${ 5 \left(\frac{1}{1-\sin x} + \frac{1}{1+\sin x}\right) = \frac{2}{\cos^2 x} }$

Our goal is to simplify this expression and understand the underlying trigonometric identities.

Understanding the Expression

The given expression involves two fractions with trigonometric functions in the denominators. The first fraction is $\frac{1}{1-\sin x}$, and the second fraction is $\frac{1}{1+\sin x}$. We need to simplify these fractions and then combine them using the given equation.

Simplifying the Fractions

To simplify the fractions, we can use the following trigonometric identities:

• $\sin^2 x + \cos^2 x = 1$
• $\sin x = \frac{1}{\csc x}$
• $\cos x = \frac{1}{\sec x}$

Using these identities, we can rewrite the fractions as follows:

$\frac{1}{1-\sin x} = \frac{1}{1-\frac{1}{\csc x}} = \frac{\csc x}{\csc x - 1}$

$\frac{1}{1+\sin x} = \frac{1}{1+\frac{1}{\csc x}} = \frac{\csc x}{\csc x + 1}$

Combining the Fractions

Now that we have simplified the fractions, we can combine them using the given equation:

$5 \left(\frac{1}{1-\sin x} + \frac{1}{1+\sin x}\right) = 5 \left(\frac{\csc x}{\csc x - 1} + \frac{\csc x}{\csc x + 1}\right)$

Using the Common Denominator

To combine the fractions, we need to find a common denominator. In this case, the common denominator is $(\csc x - 1)(\csc x + 1)$.

$5 \left(\frac{\csc x}{\csc x - 1} + \frac{\csc x}{\csc x + 1}\right) = 5 \left(\frac{\csc x(\csc x + 1) + \csc x(\csc x - 1)}{(\csc x - 1)(\csc x + 1)}\right)$

Simplifying the Numerator

Now that we have a common denominator, we can simplify the numerator:

$5 \left(\frac{\csc x(\csc x + 1) + \csc x(\csc x - 1)}{(\csc x - 1)(\csc x + 1)}\right) = 5 \left(\frac{2\csc^2 x}{\csc^2 x - 1}\right)$

Using the Pythagorean Identity

We can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression:

$5 \left(\frac{2\csc^2 x}{\csc^2 x - 1}\right) = 5 \left(\frac{2\frac{1}{\sin^2 x}}{\frac{1}{\sin^2 x} - 1}\right)$

Simplifying the Expression

Now that we have rewritten the expression, we can simplify it further:

$5 \left(\frac{2\frac{1}{\sin^2 x}}{\frac{1}{\sin^2 x} - 1}\right) = 5 \left(\frac{2}{\frac{1}{\sin^2 x} - 1}\right)$

Using the Common Denominator

To simplify the expression, we need to find a common denominator. In this case, the common denominator is $\frac{1}{\sin^2 x} - 1$.

$5 \left(\frac{2}{\frac{1}{\sin^2 x} - 1}\right) = 5 \left(\frac{2}{\frac{1 - \sin^2 x}{\sin^2 x}}\right)$

Simplifying the Expression

Now that we have a common denominator, we can simplify the expression:

$5 \left(\frac{2}{\frac{1 - \sin^2 x}{\sin^2 x}}\right) = 5 \left(\frac{2\sin^2 x}{1 - \sin^2 x}\right)$

Using the Pythagorean Identity

We can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression:

$5 \left(\frac{2\sin^2 x}{1 - \sin^2 x}\right) = 5 \left(\frac{2\sin^2 x}{\cos^2 x}\right)$

Simplifying the Expression

Now that we have rewritten the expression, we can simplify it further:

$5 \left(\frac{2\sin^2 x}{\cos^2 x}\right) = \frac{10\sin^2 x}{\cos^2 x}$

Using the Pythagorean Identity

We can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression:

$\frac{10\sin^2 x}{\cos^2 x} = \frac{10(1 - \cos^2 x)}{\cos^2 x}$

Simplifying the Expression

Now that we have rewritten the expression, we can simplify it further:

$\frac{10(1 - \cos^2 x)}{\cos^2 x} = \frac{10\sin^2 x}{\cos^2 x}$

Using the Pythagorean Identity

We can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression:

$\frac{10\sin^2 x}{\cos^2 x} = \frac{10(1 - \cos^2 x)}{\cos^2 x}$

Simplifying the Expression

Now that we have rewritten the expression, we can simplify it further:

$\frac{10(1 - \cos^2 x)}{\cos^2 x} = \frac{10\sin^2 x}{\cos^2 x}$

Conclusion

In this article, we simplified the given expression involving trigonometric functions. We used various trigonometric identities to rewrite the expression and simplify it further. The final simplified expression is $\frac{10\sin^2 x}{\cos^2 x}$. This expression can be further simplified using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$.

Final Answer

The final answer is $\boxed{\frac{10}{\cos^2 x}}$.

Introduction

In our previous article, we simplified the given expression involving trigonometric functions. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q1: What is the main trigonometric identity used in simplifying the expression?

A1: The main trigonometric identity used in simplifying the expression is the Pythagorean identity: $\sin^2 x + \cos^2 x = 1$.

Q2: How do you simplify the expression $\frac{1}{1-\sin x}$?

A2: To simplify the expression $\frac{1}{1-\sin x}$, we can use the trigonometric identity $\sin x = \frac{1}{\csc x}$. This gives us $\frac{1}{1-\sin x} = \frac{1}{1-\frac{1}{\csc x}} = \frac{\csc x}{\csc x - 1}$.

Q3: How do you combine the fractions $\frac{\csc x}{\csc x - 1}$ and $\frac{\csc x}{\csc x + 1}$?

A3: To combine the fractions, we need to find a common denominator. In this case, the common denominator is $(\csc x - 1)(\csc x + 1)$. This gives us $5 \left(\frac{\csc x}{\csc x - 1} + \frac{\csc x}{\csc x + 1}\right) = 5 \left(\frac{\csc x(\csc x + 1) + \csc x(\csc x - 1)}{(\csc x - 1)(\csc x + 1)}\right)$.

Q4: How do you simplify the numerator of the combined fraction?

A4: To simplify the numerator, we can combine the terms: $\csc x(\csc x + 1) + \csc x(\csc x - 1) = 2\csc^2 x$.

Q5: How do you simplify the expression $\frac{2\csc^2 x}{\csc^2 x - 1}$?

A5: To simplify the expression, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression: $\frac{2\csc^2 x}{\csc^2 x - 1} = \frac{2\frac{1}{\sin^2 x}}{\frac{1}{\sin^2 x} - 1}$.

Q6: How do you simplify the expression $\frac{2\frac{1}{\sin^2 x}}{\frac{1}{\sin^2 x} - 1}$?

A6: To simplify the expression, we can use the common denominator: $\frac{2\frac{1}{\sin^2 x}}{\frac{1}{\sin^2 x} - 1} = \frac{2}{\frac{1}{\sin^2 x} - 1} = \frac{2\sin^2 x}{1 - \sin^2 x}$.

Q7: How do you simplify the expression $\frac{2\sin^2 x}{1 - \sin^2 x}$?

A7: To simplify the expression, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression: $\frac{2\sin^2 x}{1 - \sin^2 x} = \frac{2\sin^2 x}{\cos^2 x}$.

Q8: How do you simplify the expression $\frac{2\sin^2 x}{\cos^2 x}$?

A8: To simplify the expression, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression: $\frac{2\sin^2 x}{\cos^2 x} = \frac{2(1 - \cos^2 x)}{\cos^2 x}$.

Q9: How do you simplify the expression $\frac{2(1 - \cos^2 x)}{\cos^2 x}$?

A9: To simplify the expression, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the expression: $\frac{2(1 - \cos^2 x)}{\cos^2 x} = \frac{2\sin^2 x}{\cos^2 x}$.

Q10: What is the final simplified expression?

A10: The final simplified expression is $\boxed{\frac{10}{\cos^2 x}}$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression involving trigonometric functions. We hope that this article has been helpful in understanding the simplification process.