Simplify The Expression: Tan ⁡ 2 X × Cos ⁡ 2 X × Sin ⁡ 2 X Tan ⁡ 2 X = 1 \tan^2 X \times \cos^2 X \times \frac{\sin^2 X}{\tan^2 X} = 1 Tan 2 X × Cos 2 X × T A N 2 X S I N 2 X ​ = 1

Introduction

In this article, we will simplify the given trigonometric expression involving tangent, cosine, and sine functions. The expression is $\tan^2 x \times \cos^2 x \times \frac{\sin^2 x}{\tan^2 x} = 1$. We will use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Understanding the Trigonometric Functions

Before we proceed with simplifying the expression, let's briefly review the trigonometric functions involved.

• Tangent (tan): The tangent of an angle $x$ is defined as the ratio of the sine of $x$ to the cosine of $x$, i.e., $\tan x = \frac{\sin x}{\cos x}$.
• Cosine (cos): The cosine of an angle $x$ is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
• Sine (sin): The sine of an angle $x$ is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Simplifying the Expression

Now, let's simplify the given expression step by step.

Step 1: Simplify the numerator

The numerator of the expression is $\tan^2 x \times \cos^2 x \times \sin^2 x$. We can simplify this by using the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$.

import sympy as sp
x = sp.symbols('x')

expr = (sp.tan(x)**2) * (sp.cos(x)**2) * (sp.sin(x)**2)

simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified expression is $\sin^4 x$.

Step 2: Simplify the denominator

The denominator of the expression is $\tan^2 x$. We can simplify this by using the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$.

import sympy as sp

x = sp.symbols('x')

expr = sp.tan(x)**2

simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified expression is $\frac{\sin^2 x}{\cos^2 x}$.

Step 3: Simplify the entire expression

Now, let's simplify the entire expression by dividing the numerator by the denominator.

import sympy as sp

x = sp.symbols('x')

expr = (sp.sin(x)**4) / ((sp.sin(x)**2) / (sp.cos(x)**2))

simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified expression is $\sin^2 x \cos^2 x$.

Conclusion

In this article, we simplified the given trigonometric expression involving tangent, cosine, and sine functions. We used various trigonometric identities and formulas to simplify the expression and arrive at the final result. The simplified expression is $\sin^2 x \cos^2 x$.

Q&A: Simplifying the Trigonometric Expression

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

Q: What is the final simplified expression?

A: The final simplified expression is $\sin^2 x \cos^2 x$.

Q: How did you simplify the expression?

A: We used various trigonometric identities and formulas to simplify the expression. We started by simplifying the numerator and denominator separately and then divided the numerator by the denominator to arrive at the final result.

Q: What is the significance of the simplified expression?

A: The simplified expression $\sin^2 x \cos^2 x$ represents the product of the sine and cosine functions squared. This expression is an important result in trigonometry and has many applications in various fields such as physics, engineering, and mathematics.

Q: Can you provide more examples of simplifying trigonometric expressions?

A: Yes, here are a few more examples of simplifying trigonometric expressions:

Example 1: Simplify the expression $\sin^2 x \cos^2 x \tan^2 x$

import sympy as sp

x = sp.symbols('x')

expr = (sp.sin(x)**2) * (sp.cos(x)**2) * (sp.tan(x)**2)

simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified expression is $\sin^4 x$.

Example 2: Simplify the expression $\cos^2 x \sin^2 x \csc^2 x$

import sympy as sp

x = sp.symbols('x')

expr = (sp.cos(x)**2) * (sp.sin(x)**2) * (sp.csc(x)**2)

simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified expression is $\sin^2 x \cos^2 x$.

Q: How can I apply the simplified expression in real-world problems?

A: The simplified expression $\sin^2 x \cos^2 x$ has many applications in various fields such as physics, engineering, and mathematics. For example, it can be used to calculate the area of a triangle, the length of a shadow, or the height of a building.

Q: Can I use the simplified expression to solve trigonometric equations?

A: Yes, the simplified expression $\sin^2 x \cos^2 x$ can be used to solve trigonometric equations. For example, you can use it to solve equations involving the sine and cosine functions.

Q: How can I verify the simplified expression?

A: You can verify the simplified expression by plugging in different values of $x$ and checking if the expression holds true. You can also use a calculator or a computer program to verify the expression.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the trigonometric expression involving tangent, cosine, and sine functions. We provided examples of simplifying trigonometric expressions and discussed the significance of the simplified expression. We also answered questions related to applying the simplified expression in real-world problems and solving trigonometric equations.