What is the Derivation of 1 + cot²A = cosec²A?

S6-SA2-0221

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The derivation of 1 + cot²A = cosec²A explains how this important trigonometric identity is proven using the basic Pythagorean identity and definitions of trigonometric ratios. It shows that the relationship between cotangent and cosecant is always true for any angle A (where cotA and cosecA are defined). This identity helps simplify complex trigonometric expressions and solve problems in geometry and physics.

Simple Example

Quick Example

Imagine you have a cricket field, and you need to calculate the distance from the wicket to a specific point on the boundary. If you know certain angles and distances, using trigonometric identities like 1 + cot²A = cosec²A can help you find unknown lengths or angles quickly, just like using a shortcut on Google Maps saves time.

Worked Example

Step-by-Step

Let's derive 1 + cot²A = cosec²A starting from the fundamental identity sin²A + cos²A = 1.

Step 1: Start with the Pythagorean identity: sin²A + cos²A = 1.

--- Step 2: To get cot²A, we need a cos²A/sin²A term. So, divide the entire equation by sin²A (assuming sinA is not 0).
(sin²A / sin²A) + (cos²A / sin²A) = (1 / sin²A)

--- Step 3: Simplify each term.
1 + (cosA/sinA)² = (1/sinA)²

--- Step 4: Recall the definitions of cotangent and cosecant.
cotA = cosA / sinA
cosecA = 1 / sinA

--- Step 5: Substitute these definitions back into the equation from Step 3.
1 + (cotA)² = (cosecA)²

--- Step 6: Write it in the standard form.
1 + cot²A = cosec²A

Answer: The identity 1 + cot²A = cosec²A is derived from sin²A + cos²A = 1 by dividing the entire equation by sin²A.

Why It Matters

Understanding trigonometric identities is crucial in fields like Engineering and Physics to design structures, predict movements, and analyze waves. For example, in Space Technology, these identities are used to calculate satellite trajectories. Careers in AI/ML also use these concepts in algorithms for image processing and data analysis.

Common Mistakes

MISTAKE: Forgetting that cot²A means (cotA)² and not cot(A²). | CORRECTION: Always remember that the '2' in cot²A applies to the entire cotA ratio, meaning cotA multiplied by cotA.

MISTAKE: Dividing the original identity sin²A + cos²A = 1 by cos²A instead of sin²A when trying to derive 1 + cot²A = cosec²A. | CORRECTION: To get cot²A (which is cos²A/sin²A), you must divide by sin²A. Dividing by cos²A would lead to tan²A + 1 = sec²A, which is a different identity.

MISTAKE: Confusing cosecA with secA or cotA with tanA in the final step. | CORRECTION: Always remember that cosecA = 1/sinA and cotA = 1/tanA = cosA/sinA. Keep the definitions clear in your mind.

Practice Questions

Try It Yourself

QUESTION: If sinA = 3/5, find the value of cosecA. | ANSWER: cosecA = 1/sinA = 1 / (3/5) = 5/3

QUESTION: Using the identity 1 + cot²A = cosec²A, if cotA = 4/3, find the value of cosecA. | ANSWER: 1 + (4/3)² = cosec²A => 1 + 16/9 = cosec²A => 25/9 = cosec²A => cosecA = sqrt(25/9) = 5/3 (assuming A is an acute angle)

QUESTION: Prove that (cosecA - cotA)(cosecA + cotA) = 1. (Hint: Use the identity derived in this lesson). | ANSWER: (cosecA - cotA)(cosecA + cotA) = cosec²A - cot²A. From the identity 1 + cot²A = cosec²A, we can rearrange to get cosec²A - cot²A = 1. Therefore, (cosecA - cotA)(cosecA + cotA) = 1.

MCQ

Quick Quiz

Which basic trigonometric identity is used as the starting point to derive 1 + cot²A = cosec²A?

sinA / cosA = tanA

sin²A + cos²A = 1

1 + tan²A = sec²A

cosecA = 1 / sinA

The Correct Answer Is:

The identity 1 + cot²A = cosec²A is derived by dividing the fundamental Pythagorean identity sin²A + cos²A = 1 by sin²A. Options A and D are definitions, and C is another derived identity.

Real World Connection

In the Real World

Imagine an engineer designing a new metro bridge in Mumbai. They need to calculate precise angles and forces. Trigonometric identities are used in complex calculations to ensure the bridge is stable and safe. Similarly, in a game like 'Free Fire' or 'BGMI', the physics engine uses trigonometry to calculate projectile trajectories when you throw a grenade or shoot, making the game realistic.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables | COTANGENT (cotA): The ratio of the adjacent side to the opposite side in a right-angled triangle, or cosA/sinA | COSECANT (cosecA): The ratio of the hypotenuse to the opposite side in a right-angled triangle, or 1/sinA | PYTHAGOREAN IDENTITY: A fundamental trigonometric identity derived from the Pythagorean theorem, like sin²A + cos²A = 1

What's Next

What to Learn Next

Great job understanding this identity! Next, you should explore the derivation of 1 + tan²A = sec²A, as it uses a similar method. Mastering these three Pythagorean identities will give you a strong foundation for solving advanced trigonometry problems and understanding their applications in real life.