Verify The Identity:${ \frac{\sin \theta + \cos \theta}{\cos \theta} - \frac{\sin \theta - \cos \theta}{\sin \theta} = \sec \theta \csc \theta }$

by ADMIN 147 views

**Verify the Identity: A Step-by-Step Guide** =====================================================

Introduction

In this article, we will delve into the world of trigonometry and explore a complex trigonometric identity involving sine and cosine functions. The given identity is:

sin⁡θ+cos⁡θcos⁡θ−sin⁡θ−cos⁡θsin⁡θ=sec⁡θcsc⁡θ\frac{\sin \theta + \cos \theta}{\cos \theta} - \frac{\sin \theta - \cos \theta}{\sin \theta} = \sec \theta \csc \theta

Our goal is to verify this identity, which means we need to prove that it is true for all values of θ\theta.

Understanding the Identity

Before we begin, let's break down the given identity and understand what it means. The left-hand side of the equation involves two fractions, each with a different denominator. The right-hand side of the equation involves the product of two trigonometric functions: sec⁡θ\sec \theta and csc⁡θ\csc \theta.

Step 1: Simplify the Left-Hand Side

To simplify the left-hand side of the equation, we need to find a common denominator for the two fractions. The common denominator is cos⁡θsin⁡θ\cos \theta \sin \theta.

import sympy as sp

theta = sp.symbols('theta')



lhs = (sp.sin(theta) + sp.cos(theta))/sp.cos(theta) - (sp.sin(theta) - sp.cos(theta))/sp.sin(theta)



simplified_lhs = sp.simplify(lhs)
print(simplified_lhs)

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation involves the product of two trigonometric functions: sec⁡θ\sec \theta and csc⁡θ\csc \theta. We can simplify this expression by using the definitions of sec⁡θ\sec \theta and csc⁡θ\csc \theta.

# Define the right-hand side of the equation
rhs = sp.sec(theta)*sp.csc(theta)


simplified_rhs = sp.simplify(rhs)
print(simplified_rhs)

Step 3: Verify the Identity

Now that we have simplified both sides of the equation, we can verify the identity by checking if the two expressions are equal.

# Verify the identity
identity_verified = sp.simplify(simplified_lhs - simplified_rhs) == 0
print(identity_verified)

Conclusion

In this article, we have verified the given trigonometric identity involving sine and cosine functions. We simplified both sides of the equation and checked if they are equal. The result confirms that the identity is true for all values of θ\theta.

Q&A

Q: What is the purpose of verifying trigonometric identities?

A: Verifying trigonometric identities is an essential step in mathematics, as it helps us understand the relationships between different trigonometric functions and ensures that our calculations are correct.

Q: How do I simplify trigonometric expressions?

A: To simplify trigonometric expressions, you can use the following steps:

  1. Use the definitions of the trigonometric functions.
  2. Simplify the expression using algebraic manipulations.
  3. Use trigonometric identities to simplify the expression further.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Q: How do I use Python to verify trigonometric identities?

A: To verify trigonometric identities using Python, you can use the sympy library, which provides a powerful tool for symbolic mathematics.

import sympy as sp


theta = sp.symbols('theta')



lhs = (sp.sin(theta) + sp.cos(theta))/sp.cos(theta) - (sp.sin(theta) - sp.cos(theta))/sp.sin(theta)



simplified_lhs = sp.simplify(lhs)



rhs = sp.sec(theta)*sp.csc(theta)



simplified_rhs = sp.simplify(rhs)



identity_verified = sp.simplify(simplified_lhs - simplified_rhs) == 0
print(identity_verified)

Q: What are some real-world applications of trigonometry?

A: Trigonometry has numerous real-world applications, including:

  • Navigation: Trigonometry is used in navigation to calculate distances and directions.
  • Physics: Trigonometry is used in physics to describe the motion of objects.
  • Engineering: Trigonometry is used in engineering to design and build structures.
  • Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.