Secant Square X Formula

Secant is one of the functions of Trigonometry which are applied on the right-angled triangle along with sine, cosine, tangent, cosecant, and cotangent and it can be derived by dividing the hypotenuse with the adjacent side of the triangle. Though the secant function is not used much frequently, it is still useful to understand the other Trigonometric functions. In this article, we will learn about the formula of sec square x along with its derivation and some examples. 

You will learn how it can be derived and used in Trigonometry and with this how can you solve the advanced examples of trigonometry chapters. 

Introduction 

Trigonometry functions are used on a right-angled triangle and in a right-angled triangle. Secant is considered as the value of an angle which is derived when we divide the length of the hypotenuse with the length of its adjacent side. It is abbreviated as "sec" in formulas of trigonometry and can be derived as a reciprocal of "cosine". In trigonometry, the most common functions which are used are sine, cosine and tangent whereas the other three are rarely used that are cosecant, secant and cotangent and sometimes, even sometimes most of the calculators do not include buttons for these three functions and also you will find them missing in software functions libraries as well. Secant can be written as follows:

Sec x = H/A

Where H is hypotenuse & A is an adjacent side.

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Sec Square x Formula

If we talk about the secant, then in trigonometric functions they are usually involved in the form of squares in expressions as well as equations. We can simplify the equations as well expressions only by transferring the functions of the secant square into some equivalent form. Thus, it becomes more important in order to learn about the secant square functions so that other advanced functions of Trigonometry can also be understood. It is mostly used in two cases such as when the secant square function is equal to the summation of one as well as tangent square function and on the other hand, we also can say that the sum of tangent square function and one is also simplified as secant square function. 

The formula of secant square function or we can say sec square x formula can be written in the following two forms which are mentioned below. The formula of sec square x in mathematics can be written in any of these two forms:

• Sec²x = 1 + tan²x

• Sec²A = 1 + tan²A

Formula Proof

Let suppose that "theta" is a symbol that represents the angles of a right-angled triangle. In Trigonometry, secant, as well as a tangent, can be written in the form of sec theta and tan theta respectively. According to the Pythagorean identity of secant as well as tangent functions, in trigonometry secant and tangent functions can be written in the following form:

Sec²θ - tan²θ = 1

Therefore, Sec²θ = 1 + tan²θ

Thus, this is how the secant square x formula can be derived and proved. Through this, we can say that the secant square function is equal to the summation of one as well as the tangent square function.

Examples

Let's solve some examples with the help of the secant square x formula which are given below:

Example 1: What is the value of sec square x, if tan x = 2/5?

Solution: Sec²x = 1 + tan²x

= 1 + (2/5)²

= 1 + 4/25

= (25 + 4) / 25

= 29/25

Hence, Sec²x = 29/25

Example 2: What is the value of Sec square x, if tan x = 1/3?

Solution: Sec²x = 1 + tan²x

= 1 + (4/5)²

= 1 + 1/9

= (10) / 9

Hence, Sec²x = 10/9

Example 3: find the value of sec²x, if the value of cot x = 2/5  

Solution:

Given, cot x = 2/5

tan x = 1/cot x = 5/2

sec²x = 1 + tan²x

= 1 + (5/2)²

= 1 + 25/4

= 29/4

Example 4: Simplify:  (1 + tan²A) (1 – sin²A) 

Solution:

= (1 + tan²A)(1 – sin²A)

= sec²A . cos²A

= (1/cos²A) . cos²A

= 1

Conclusion

Thus, to sum the discussion of the sec square x formula, we can say that it can be easily derived from the Pythagorean functions of mathematics and it is equal to the sum of tangent square functions and one. It can be directly used where such expression you can see in the questions or otherwise such expressions can be made in order to reach this form so that formula of sec square x, can easily be used in order to reach the final answer for example in one of the examples, we find out the value of tan x from cot x in order to reach sec²x value.