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What is the Identity 1 + cot²A = cosec²A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity 1 + cot²A = cosec²A is a fundamental relationship in trigonometry that connects the cotangent and cosecant ratios for any angle A. It is derived from the Pythagorean identity sin²A + cos²A = 1, by dividing all terms by sin²A. This identity helps simplify complex trigonometric expressions and solve problems.
Simple Example
Quick Example
Imagine you're checking cricket scores, and you know the 'cot' value of a player's strike rate (cot A). If you add 1 to the square of this cot value, you will always get the square of the 'cosec' value for that same strike rate (cosec A). It's a fixed mathematical rule, just like knowing 2+2 always equals 4.
Worked Example
Step-by-Step
Let's verify the identity 1 + cot²A = cosec²A for a specific angle, say A = 30 degrees.
Step 1: Find cot(30 degrees).
We know tan(30 degrees) = 1/sqrt(3).
So, cot(30 degrees) = 1 / tan(30 degrees) = 1 / (1/sqrt(3)) = sqrt(3).
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Step 2: Calculate cot²(30 degrees).
cot²(30 degrees) = (sqrt(3))² = 3.
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Step 3: Calculate 1 + cot²(30 degrees).
1 + cot²(30 degrees) = 1 + 3 = 4.
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Step 4: Find sin(30 degrees).
We know sin(30 degrees) = 1/2.
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Step 5: Find cosec(30 degrees).
cosec(30 degrees) = 1 / sin(30 degrees) = 1 / (1/2) = 2.
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Step 6: Calculate cosec²(30 degrees).
cosec²(30 degrees) = (2)² = 4.
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Step 7: Compare the results.
From Step 3, 1 + cot²(30 degrees) = 4.
From Step 6, cosec²(30 degrees) = 4.
Since both sides are equal, the identity 1 + cot²A = cosec²A is verified for A = 30 degrees.
Answer: Verified, as both sides equal 4.
Why It Matters
Trigonometric identities are crucial in fields like Physics for calculating projectile motion or wave patterns, and in Engineering for designing bridges or satellite orbits. Engineers use these to ensure structures are stable, and scientists in Space Technology use them to track spacecraft trajectories. Knowing these identities can open doors to careers in ISRO or DRDO.
Common Mistakes
MISTAKE: Writing 1 + cot A = cosec A (forgetting the squares) | CORRECTION: Always remember the squares on cot A and cosec A. The identity is 1 + cot²A = cosec²A.
MISTAKE: Confusing cot²A with tan²A, or cosec²A with sec²A. | CORRECTION: Remember that cotangent is related to cosecant, and tangent is related to secant. The identities are 1 + cot²A = cosec²A and 1 + tan²A = sec²A.
MISTAKE: Assuming the identity only works for specific angles like 30 or 45 degrees. | CORRECTION: This is an identity, meaning it holds true for ALL valid angles A, not just specific ones. It's a universal rule in trigonometry.
Practice Questions
Try It Yourself
QUESTION: If cot A = 4/3, find cosec A using the identity 1 + cot²A = cosec²A. | ANSWER: cosec A = 5/3
QUESTION: Simplify the expression (cosec²A - cot²A) * (1 + cot²A). | ANSWER: cosec²A
QUESTION: Prove that (1 + cot A)² + (1 - cot A)² = 2 cosec²A. | ANSWER: (1 + 2cotA + cot²A) + (1 - 2cotA + cot²A) = 2 + 2cot²A = 2(1 + cot²A) = 2cosec²A. Hence Proved.
MCQ
Quick Quiz
Which of the following expressions is equivalent to cosec²A - 1?
tan²A
cot²A
sec²A
sin²A
The Correct Answer Is:
B
The identity is 1 + cot²A = cosec²A. Rearranging it, we get cot²A = cosec²A - 1. Therefore, option B is correct.
Real World Connection
In the Real World
Imagine an engineer designing a flyover in Mumbai. They use trigonometry to calculate angles and forces. When simplifying complex equations involving angles, identities like 1 + cot²A = cosec²A help them make calculations faster and more accurate. This ensures the flyover is safe and strong for all the vehicles, from auto-rickshaws to lorries.
Key Vocabulary
Key Terms
Identity: An equation that is true for all valid values of the variables involved. | Cotangent (cot A): The ratio of the adjacent side to the opposite side in a right-angled triangle, or 1/tan A. | Cosecant (cosec A): The ratio of the hypotenuse to the opposite side in a right-angled triangle, or 1/sin A. | Trigonometry: The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.
What's Next
What to Learn Next
Great job understanding this identity! Next, you should explore the identity 1 + tan²A = sec²A. It's very similar and will help you master all the fundamental trigonometric identities, which are essential building blocks for advanced trigonometry problems.
