Prove The Identity.${ \frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan X }$

Mar 12, 2025 by ADMIN 92 views

Proving the Identity: $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$

In this article, we will delve into the world of trigonometric identities and explore the proof of a specific identity involving trigonometric functions. The identity in question is $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$ . We will break down the proof into manageable steps, using various trigonometric identities and properties to arrive at the final result.

Understanding the Trigonometric Functions

Before we begin the proof, it's essential to understand the trigonometric functions involved. The secant, cosecant, sine, and cosine functions are reciprocal functions of each other. The secant and cosecant functions are defined as:

$\sec x = \frac{1}{\cos x}$
$\csc x = \frac{1}{\sin x}$

The sine and cosine functions are defined as:

$\sin x = \frac{y}{r}$
$\cos x = \frac{x}{r}$

where $r$ is the radius of the unit circle and $y$ is the $y$ -coordinate of the point on the unit circle corresponding to the angle $x$ .

To prove the identity, we will start by simplifying the left-hand side of the equation.

Step 1: Simplify the Left-Hand Side

We can simplify the left-hand side of the equation by using the definitions of the secant and cosecant functions.

$\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}$

Using the definitions of the secant and cosecant functions, we can rewrite the equation as:

$\frac{\frac{1}{\cos (-x)}}{\frac{1}{\sin (-x)}}+\frac{\sin (-x)}{\cos (-x)}$

Simplifying the equation further, we get:

$\frac{\sin (-x)}{\cos (-x)}+\frac{\sin (-x)}{\cos (-x)}$

Step 2: Combine Like Terms

We can combine the two like terms on the left-hand side of the equation.

$\frac{\sin (-x)}{\cos (-x)}+\frac{\sin (-x)}{\cos (-x)}$

Combining the two terms, we get:

$\frac{2 \sin (-x)}{\cos (-x)}$

Step 3: Simplify the Expression

We can simplify the expression further by using the properties of the sine and cosine functions.

$\frac{2 \sin (-x)}{\cos (-x)}$

Using the property that $\sin (-x) = -\sin x$ , we can rewrite the equation as:

$\frac{-2 \sin x}{\cos (-x)}$

Step 4: Simplify the Cosine Function

We can simplify the cosine function by using the property that $\cos (-x) = \cos x$ .

$\frac{-2 \sin x}{\cos (-x)}$

Using the property that $\cos (-x) = \cos x$ , we can rewrite the equation as:

$\frac{-2 \sin x}{\cos x}$

Step 5: Simplify the Expression

We can simplify the expression further by using the definition of the tangent function.

$\frac{-2 \sin x}{\cos x}$

Using the definition of the tangent function, we can rewrite the equation as:

$-2 \tan x$

Conclusion

In this article, we have proven the identity $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$ using various trigonometric identities and properties. We have broken down the proof into manageable steps, simplifying the left-hand side of the equation and using the properties of the sine and cosine functions to arrive at the final result.

The secant and cosecant functions are reciprocal functions of each other.
The sine and cosine functions are defined as $\sin x = \frac{y}{r}$ and $\cos x = \frac{x}{r}$ , respectively.
The tangent function is defined as $\tan x = \frac{\sin x}{\cos x}$ .
The identity $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$ can be proven using various trigonometric identities and properties.

In conclusion, the proof of the identity $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$ is a great example of how trigonometric identities and properties can be used to simplify complex expressions and arrive at a final result. By breaking down the proof into manageable steps and using various trigonometric identities and properties, we have been able to prove the identity and gain a deeper understanding of the trigonometric functions involved.
Q&A: Proving the Identity $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$

In our previous article, we proved the identity $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$ using various trigonometric identities and properties. In this article, we will answer some common questions that readers may have about the proof and the identity itself.

Q: What is the significance of the negative sign in the identity?

A: The negative sign in the identity is due to the fact that we are dealing with negative angles. When we substitute $-x$ into the trigonometric functions, we get negative values. The negative sign is simply a result of the properties of the trigonometric functions.

Q: Can you explain the concept of reciprocal functions?

A: Yes, certainly. Reciprocal functions are functions that are equal to the reciprocal of another function. In the case of the secant and cosecant functions, they are reciprocal functions of each other. The secant function is equal to the reciprocal of the cosine function, and the cosecant function is equal to the reciprocal of the sine function.

Q: How do you simplify the left-hand side of the equation?

A: To simplify the left-hand side of the equation, we use the definitions of the secant and cosecant functions. We rewrite the equation using the definitions of these functions and then simplify the expression.

Q: Can you explain the concept of the tangent function?

A: Yes, certainly. The tangent function is defined as the ratio of the sine function to the cosine function. It is equal to $\tan x = \frac{\sin x}{\cos x}$ .

Q: How do you use the properties of the sine and cosine functions to simplify the expression?

A: We use the properties of the sine and cosine functions to simplify the expression by rewriting the equation using the definitions of these functions. We then use the properties of the sine and cosine functions to simplify the expression further.

Q: Can you explain the concept of the unit circle?

A: Yes, certainly. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions in terms of the coordinates of a point on the circle.

Q: How do you use the unit circle to define the trigonometric functions?

A: We use the unit circle to define the trigonometric functions by considering the coordinates of a point on the circle. The sine function is defined as the ratio of the $y$ -coordinate to the radius, and the cosine function is defined as the ratio of the $x$ -coordinate to the radius.

Q: Can you explain the concept of the reciprocal identity?

A: Yes, certainly. The reciprocal identity is an identity that relates the reciprocal of a function to the function itself. In the case of the secant and cosecant functions, the reciprocal identity is $\sec x = \frac{1}{\cos x}$ and $\csc x = \frac{1}{\sin x}$ .

Q: How do you use the reciprocal identity to simplify the expression?

A: We use the reciprocal identity to simplify the expression by rewriting the equation using the definitions of the secant and cosecant functions. We then use the properties of the sine and cosine functions to simplify the expression further.

Conclusion

In this article, we have answered some common questions that readers may have about the proof and the identity $\frac{\sec (-x)}{\csc (-x)}+\frac{\sin (-x)}{\cos (-x)}=-2 \tan x$ . We have explained the concept of reciprocal functions, the unit circle, and the tangent function, and we have used these concepts to simplify the expression and arrive at the final result.

The negative sign in the identity is due to the fact that we are dealing with negative angles.
Reciprocal functions are functions that are equal to the reciprocal of another function.
The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.
The tangent function is defined as the ratio of the sine function to the cosine function.
The reciprocal identity is an identity that relates the reciprocal of a function to the function itself.