# Secant Square x Formula

Trigonometry is one of the most significant topics of mathematics, with numerous applications in a wide range of fields. The study of the connection between the sides and angles of a right-angle triangle is the main focus of trigonometry. As a result, using trigonometric formulae, functions, or trigonometric identities, it is possible to find the missing or unknown angles or sides of a right triangle. Angles in trigonometry can be measured in degrees or radians.

**Secant Trigonometric Ratio**

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

**Secant Square x Formula**

The secant square x ratio is denoted by the abbreviation sec^{2} 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 sec^{2} x is π. Its formula is equivalent to the sum of unity and the tangent square function.

**Formula **

sec^{2}x = 1 + tan^{2}xwhere,

x is one of the angles of the right triangle,

tan x is the tangent ratio for angle x.

**Derivation**

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

We know, sin

^{2}x + cos^{2}x = 1.Dividing both sides by cos

^{2}x, we get(sin

^{2}x/cos^{2}x) + (cos^{2}x/cos^{2}x) = 1/cos^{2}xtan

^{2}x + 1 = sec^{2}x=> sec

^{2}x = 1 + tan^{2}xThis derives the formula for secant square x ratio.

**Sample Problems**

**Problem 1. If tan x = 3/4, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, tan x = 3/4.

Using the formula we get,

sec

^{2}x = 1 + tan^{2}x= 1 + (3/4)

^{2}= 1 + 9/16

= 25/16

**Problem 2. If tan x = 12/5, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, tan x = 12/5.

Using the formula we get,

sec

^{2}x = 1 + tan^{2}x= 1 + (12/5)

^{2}= 1 + 144/25

= 169/25

**Problem 3. If sin x = 8/17, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, sin x = 8/17.

Find the value of cos x using the formula sin

^{2}x + cos^{2}x = 1.cos x = √(1 – (64/289))

= √(225/289)

= 15/17

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

Using the formula we get,

sec

^{2}x = 1 + tan^{2}x= 1 + (8/15)

^{2}= 1 + 64/225

= 289/225

**Problem 4. If cot x = 8/15, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, cot x = 8/15.

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

Using the formula we get,

sec

^{2}x = 1 + tan^{2}x= 1 + (15/8)

^{2}= 1 + 225/64

= 289/64

**Problem 5. If cos x = 12/13, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, cos x = 12/13.

Find the value of sin x using the formula sin

^{2}x + cos^{2}x = 1.sin x = √(1 – (144/169))

= √(25/169)

= 5/13

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

Using the formula we get,

sec

^{2}x = 1 + tan^{2}x= 1 + (5/12)

^{2}= 1 + 25/144

= 169/144

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