Trigonometry is a branch of mathematics that deals with the relationship between the lengths and angles of triangles. Trigonometric identities play a crucial role in simplifying trigonometric expressions and equations. In this article, we will explore and prove the trigonometry identity:
[ \frac{1 + \sec(A)}{\sec(A)} = \frac{\sin^2 A}{1 – \cos A} ]
This identity involves the secant, sine, and cosine functions. To prove this identity, we will employ various trigonometric relationships, properties, and formulas.
Understanding the Components
Before we delve into proving the identity, let’s familiarize ourselves with the trigonometric functions involved:
 Secant (sec): The secant of an angle in a right triangle is the ratio of the hypotenuse to the adjacent side.
 Sine (sin): The sine of an angle in a right triangle is the ratio of the length of the side opposite the given angle to the hypotenuse.
 Cosine (cos): The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
Proof of Trigonometry Identity
To prove the given identity [ \frac{1 + \sec(A)}{\sec(A)} = \frac{\sin^2 A}{1 – \cos A} ], we will start by expressing secant in terms of sine and cosine using the fundamental identity:
[ \sec(A) = \frac{1}{\cos(A)} ]
Substitute this into the identity and simplify the expression on both sides:
[ \frac{1 + \frac{1}{\cos(A)}}{\frac{1}{\cos(A)}} = \frac{\sin^2 A}{1 – \cos A} ]
[ \frac{\cos(A) + 1}{1} = \frac{\sin^2 A}{1 – \cos A} ]
[ \cos(A) + 1 = \frac{\sin^2 A}{1 – \cos A} ]
Now, express the sine in terms of cosine using the Pythagorean identity [ \sin^2 A + \cos^2 A = 1 ]:
[ 1 – \cos^2 A + 1 = \frac{\sin^2 A}{1 – \cos A} ]
[ 2 – \cos^2 A = \frac{\sin^2 A}{1 – \cos A} ]
[ \cos^2 A – 2 = \frac{\sin^2 A}{\cos A – 1} ]
Using the Pythagorean identity again, we have [ \cos^2 A = 1 – \sin^2 A ]:
[ 1 – \sin^2 A – 2 = \frac{\sin^2 A}{1 – \cos A} ]
[ \sin^2 A – 1 = \frac{\sin^2 A}{1 – \cos A} ]
[ \frac{\sin^2 A – 1}{1} = \frac{\sin^2 A}{1 – \cos A} ]
[ \sin^2 A + 1 = \frac{\sin^2 A}{1 – \cos A} ]
[ 1 + \sin^2 A = \frac{\sin^2 A}{1 – \cos A} ]
[ \frac{1 + \sin^2 A}{\sin^2 A} = \frac{\sin^2 A}{1 – \cos A} ]
[ \frac{1}{\sin^2 A} + \frac{\sin^2 A}{\sin^2 A} = \frac{\sin^2 A}{1 – \cos A} ]
[ \csc^2 A + 1 = \frac{\sin^2 A}{1 – \cos A} ]
[ \frac{1}{\cos^2 A} + 1 = \frac{\sin^2 A}{1 – \cos A} ]
[ \sec^2 A = \frac{\sin^2 A}{1 – \cos A} ]
Application and Verification
The derived expression [ \sec^2 A = \frac{\sin^2 A}{1 – \cos A} ] is a valid trigonometric identity. This identity can be used in various trigonometric calculations, proofs, and problemsolving scenarios. By applying the properties of trigonometric functions and identities, this relationship can simplify complex trigonometric expressions and equations.
To further verify the identity, you can substitute different angles A into the expression and calculate the values on both sides. By ensuring that both sides of the equation yield the same results, the validity of the identity can be confirmed.
Frequently Asked Questions (FAQs)

What are trigonometric identities?
Trigonometric identities are mathematical equations that involve trigonometric functions and hold true for all values of the variables within their domains. 
How are trigonometric identities used in mathematics?
Trigonometric identities are used to simplify trigonometric expressions, prove mathematical theorems, solve equations, and analyze problems in various fields such as physics, engineering, and computer science. 
What is the Pythagorean identity in trigonometry?
The Pythagorean identity in trigonometry states that the sum of the squares of the sine and cosine of an angle is equal to 1: [ \sin^2 A + \cos^2 A = 1 ]. 
Why are trigonometric identities important?
Trigonometric identities are essential tools in trigonometry as they help establish relationships between trigonometric functions, facilitate calculations, and simplify complex expressions, making mathematical tasks more manageable. 
Can trigonometric identities be derived from basic trigonometric functions?
Yes, many trigonometric identities can be derived from fundamental trigonometric functions such as sine, cosine, and tangent using algebraic manipulations, geometric interpretations, and trigonometric properties. 
What is the difference between a trigonometric equation and a trigonometric identity?
A trigonometric equation is an equation involving trigonometric functions that is true for specific values of the variables, whereas a trigonometric identity is an equation that holds true for all values of the variables within their domains. 
How can trigonometric identities help simplify mathematical calculations?
By using trigonometric identities, complex trigonometric expressions can be simplified into more manageable forms, allowing for easier calculations, problemsolving, and analysis in various mathematical contexts. 
Can trigonometric identities be applied in reallife scenarios?
Yes, trigonometric identities have practical applications in fields such as architecture, physics, engineering, astronomy, computer graphics, and more. They are used to model and solve realworld problems involving angles, distances, velocities, and oscillations. 
What are some common trigonometric identities every student should know?
Some common trigonometric identities include the Pythagorean identities, reciprocal identities, quotient identities, cofunction identities, sum and difference identities, double angle identities, and halfangle identities. 
How can I improve my understanding of trigonometric identities and their applications?
To enhance your knowledge of trigonometric identities, practice solving trigonometric problems, work on proofs, explore applications in different fields, and seek clarification from textbooks, online resources, and educators. Building a strong foundation in trigonometry will aid in mastering identities and their practical use.
In conclusion, trigonometric identities such as [ \frac{1 + \sec(A)}{\sec(A)} = \frac{\sin^2 A}{1 – \cos A} ] are powerful tools that enable us to manipulate trigonometric expressions, establish connections between different functions, and solve complex mathematical problems. By understanding the relationships between trigonometric functions and identities, we can enhance our problemsolving skills and mathematical proficiency.