Secant Square x Formula

Secant is one of the six fundamental identities in trigonometry, specifically a Pythagorean identity. There are various identities involving secant, one of which is the secant square x formula. This formula is expressed as:

sec² x = 1 + tan² x

This formula represents the relationship between the secant and tangent functions. In this article, we will discuss this formula in detail, including its proof as well as some solved examples related to it.

The ratio of the lengths of any two sides of a right triangle is called a trigonometric ratio. In trigonometry, these ratios link the ratio of sides of a right triangle to the angle. The secant ratio is expressed as the ratio of the hypotenuse (longest side) to the side corresponding to a given angle in a right triangle.

It is the reciprocal of the cosine ratio and is denoted by the abbreviation sec.

If θ is the angle that lies between the base and hypotenuse of a right-angled triangle then,

sec θ = Hypotenuse/Base = 1/cos θ

Here, hypotenuse is the longest side of right triangle and base is the side adjacent to the angle.

The secant square x ratio is denoted by the abbreviation sec² x. It’s a trigonometric function that returns the square of the secant function value for an angle x. The period of the function sec x is 2π, but the period of sec² x is π. Its formula is equivalent to the sum of unity and the tangent square function.

sec² x = 1 + tan² x

Where,

• x is one of the angles of the right triangle,
• tan x is the tangent ratio for angle x.

Derivation of Secant Square Formula

The formula for secant square x is derived by using the identity of sum of squares of sine and cosine ratios.

We know, sin² x + cos² x = 1.

Dividing both sides by cos² x, we get

(sin² x/cos² x) + (cos² x/cos² x) = 1/cos² x

tan² x + 1 = sec² x

sec² x = 1 + tan² x

This derives the formula for secant square x ratio.

Read More,

• Trigonometry
• Trigonometric Formulas
• Trigonometric Identities
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Problem 1. If tan x = 3/4, find the value of sec² x using the formula.

Solution:

We have, tan x = 3/4.

Using the formula we get,

sec² x = 1 + tan² x

⇒ sec² x = 1 + (3/4)²

⇒ sec² x = 1 + 9/16

⇒ sec² x = 25/16

Problem 2. If tan x = 12/5, find the value of sec² x using the formula.

Solution:

We have, tan x = 12/5.

Using the formula we get,

sec² x = 1 + tan² x

⇒ sec² x = 1 + (12/5)²

⇒ sec² x = 1 + 144/25

⇒ sec² x = 169/25

Problem 3. If sin x = 8/17, find the value of sec² x using the formula.

Solution:

We have, sin x = 8/17.

Find the value of cos x using the formula sin² x + cos² x = 1.

cos x = √(1 – (64/289))

⇒ cos x = √(225/289)

⇒ cos x = 15/17

So, tan x = sin x/cos x = 8/15

Using the formula we get,

sec² x = 1 + tan² x

⇒ sec² x = 1 + (8/15)²

⇒ sec² x = 1 + 64/225

⇒ sec² x = 289/225

Problem 4. If cot x = 8/15, find the value of sec² x using the formula.

Solution:

We have, cot x = 8/15.

So, tan x = 1/cot x = 15/8

Using the formula we get,

sec² x = 1 + tan² x

⇒ sec² x = 1 + (15/8)²

⇒ sec² x = 1 + 225/64

⇒ sec² x = 289/64

Problem 5. If cos x = 12/13, find the value of sec² x using the formula.

Solution:

We have, cos x = 12/13.

Find the value of sin x using the formula sin² x + cos² x = 1.

sin x = √(1 – (144/169))

⇒ sin x = √(25/169)

⇒ sin x = 5/13

So, tan x = sin x/cos x = 5/12

Using the formula we get,

sec² x = 1 + tan² x

⇒ sec² x = 1 + (5/12)²

⇒ sec² x = 1 + 25/144

⇒ sec² x = 169/144

What is the Secant Square Formula?

The Secant Square Formula is a trigonometric identity that states:

sec² x = 1 + tan² x

How is the Secant Square Formula derived?

The formula can be derived from the Pythagorean identity:

sin² x + cos² x = 1

Where is the Secant Square Formula used?

The Secant Square Formula is frequently used in calculus, particularly in integration and differentiation of trigonometric functions.

Can the Secant Square Formula be applied to all angles?

Yes, the Secant Square Formula is valid for all angles θ where cos⁡ θ ≠ 0.