Cosec Cot Formula

Cosec cot formula is one of the fundamental trigonometric identities that is used for various purposes. This identity states that 1 + cot² θ = cosec²θ, and is also called cosec cot equation.

In this article, we, have covered Cosec Cot Formula, Derivation of Cosec Cot Formula, Examples related to Cosec Cot formula and others in detail.

Cosec Cot formula is a Pythagorean Pythagoras’ in trigonometry since it is based on Pythagoras’ theorem. It says that at every angle, the square of cosecant is equal to the sum of the square of cotangent and unity.

cosec² θ – cot² θ = 1

Image added below shows the cosec cot formula:

Consider a right triangle ABC with angle θ between its base (BC) and hypotenuse(AC).

Applying Pythagoras’ theorem on this triangle, we get

AC² = AB² + BC²            

Dividing both sides by AB², we get

AC²/AB² = AB²/AB² + BC²/AB²

(AC/AB)² = 1 + (BC/AB)²    ……. (1)

We know, for angle θ,

cosec θ = Hypotenuse/Perpendicular

cosec θ = AC/AB                ……. (2)

Also, we have

cot θ = Base/Perpendicular

cot θ = BC/AB                   ……. (3)

Using (2) and (3) in (1), we get

cosec² θ = 1 + cot² θ

This proves the cosec cot formula.

Article Related to Cosec Cot Formula:

• Trigonometric Table
• Trigonometric Identities
• Trigonometry in Maths
• Trigonometric Ratios

Problem 1. If cot θ = 3/4, find the value of cosec θ using the formula.

Solution:

We have,

cot θ = 3/4

Using the formula we have,

cosec² θ = 1 + cot² θ

cosec² θ = 1 + (3/4)²

cosec² θ = 1 + 9/16

cosec² θ = 25/16

cosec θ = 5/4

Problem 2. If cot θ = 12/5, find the value of cosec θ using the formula.

Solution:

We have,

cot θ = 12/5

Using the formula we have,

cosec² θ = 1 + cot² θ

cosec² θ = 1 + (12/5)²

cosec² θ = 1 + 144/25

cosec² θ = 169/25

cosec θ = 13/5

Problem 3. If cos θ = 4/5, find the value of cosec θ using the formula.

Solution:

We have, cos θ = 4/5.

Clearly sin θ = 3/5. Hence we have, cot θ = 4/3.

Using the formula we have,

cosec² θ = 1 + cot² θ

cosec² θ = 1 + (4/3)²

cosec² θ = 1 + 16/9

cosec² θ = 25/9

cosec θ = 5/3

Problem 4. If sin θ = 12/13, find the value of cosec θ using the formula.

Solution:

We have, sin θ = 12/13.

Clearly cos θ = 5/13. Hence we have, cot θ = 12/5.

Using the formula we have,

cosec² θ = 1 + cot² θ

cosec² θ = 1 + (12/5)²

cosec² θ = 1 + 144/25

cosec² θ = 169/25

cosec θ = 13/5

Problem 5. If sin θ = 4/5, find the value of cot θ using the formula.

Solution:

We have, sin θ = 4/5.

Clearly cosec θ = 5/4.

Using the formula we have,

cosec² θ = 1 + cot² θ

cot² θ = (5/4)² – 1

cot² θ = 25/16 – 1

cot² θ = 9/16

cot θ = 3/4

Problem 6. If sec θ = 17/8, find the value of cosec θ using the formula.

Solution:

We have, sec θ = 17/8.

Clearly cos θ = 8/17. Hence we have, cot θ = 8/15.

Using the formula we have,

cosec² θ = 1 + cot² θ

cosec² θ = 1 + (8/15)²

cosec² θ = 1 + 64/225

cosec² θ = 289/225

cosec θ = 17/15

Problem 7. If tan θ = 12/5, find the value of cosec θ using the formula.

Solution:

We have, tan θ = 12/5.

Hence we have, cot θ = 5/12.

Using the formula we have,

cosec² θ = 1 + cot² θ

cosec² θ = 1 + (5/12)²

cosec² θ = 1 + 25/144

cosec² θ = 169/144

cosec θ = 13/12

Q1. Find the value of cosec 30° and cot 45°.

Q2. If sin θ = 3/5, find cosec θ and cot θ.

Q3. Determine the value of cosec α and cot α if tan α = 5/12.

Q4. Given that cot β = 7/24, find cosec β.

Q5. If cot γ = -2, calculate cosec γ.

What is the Equation for Cot and Cosec?

Cot x is defined as: cot x = cos x/sin x whereas, Cosec x is defined as: cosec x = 1/sin x.

What is csc cot Formula?

Csc cot formula is a trigonometric identity and is defined as: csc² θ = 1 + cot² θ.

What is the Formula for the Cosecx?

Cosec x is the reciprocal of sin x and is defined as: cosec x = 1/sin x.