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

Feb 25, 2025 by ADMIN 85 views

**Simplify the Expression: A Step-by-Step Guide** =====================================================

Introduction

In this article, we will simplify the given expression: $\left(\tan x+\frac{\cos x}{\sin x}\right) \sin ^2 x=\tan x$ . This expression involves trigonometric functions, and we will use various identities and properties to simplify it.

Understanding the Expression

The given expression is a product of two terms: $\left(\tan x+\frac{\cos x}{\sin x}\right)$ and $\sin ^2 x$ . We need to simplify this expression to make it easier to work with.

Step 1: Simplify the First Term

Let's start by simplifying the first term: $\left(\tan x+\frac{\cos x}{\sin x}\right)$ . We can rewrite $\tan x$ as $\frac{\sin x}{\cos x}$ .

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expression
expr = (sp.tan(x) + sp.cos(x)/sp.sin(x))

# Simplify the expression
simplified_expr = sp.simplify(expr)
print(simplified_expr)

Step 2: Simplify the Second Term

Now, let's simplify the second term: $\sin ^2 x$ . We can rewrite this as $(\sin x)^2$ .

# Define the expression
expr = (sp.sin(x))**2
print(expr)

Step 3: Multiply the Two Terms

Now that we have simplified both terms, we can multiply them together.

# Define the expressions
expr1 = simplified_expr
expr2 = (sp.sin(x))**2

# Multiply the expressions
result = expr1 * expr2
print(result)

Step 4: Simplify the Result

Finally, let's simplify the result.

# Simplify the result
simplified_result = sp.simplify(result)
print(simplified_result)

Conclusion

In this article, we simplified the given expression: $\left(\tan x+\frac{\cos x}{\sin x}\right) \sin ^2 x=\tan x$ . We used various identities and properties to simplify the expression, and we arrived at the final result.

Q&A

Q: What is the final simplified expression? A: The final simplified expression is $\sin ^2 x \tan x + \cos x$ .

Q: How did you simplify the first term? A: We simplified the first term by rewriting $\tan x$ as $\frac{\sin x}{\cos x}$ .

Q: How did you simplify the second term? A: We simplified the second term by rewriting $\sin ^2 x$ as $(\sin x)^2$ .

Q: What is the purpose of multiplying the two terms? A: We multiplied the two terms together to simplify the expression.

Q: How did you simplify the result? A: We simplified the result by using the simplify function from the SymPy library.

Frequently Asked Questions

Q: What is the difference between $\tan x$ and $\frac{\sin x}{\cos x}$ ? A: $\tan x$ and $\frac{\sin x}{\cos x}$ are equivalent expressions.

Q: What is the difference between $\sin ^2 x$ and $(\sin x)^2$ ? A: $\sin ^2 x$ and $(\sin x)^2$ are equivalent expressions.

Q: Why did you use the simplify function from the SymPy library? A: We used the simplify function from the SymPy library to simplify the expression.