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What is the Identity 1 + tan^2 A = sec^2 A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity 1 + tan^2 A = sec^2 A is a fundamental relationship in trigonometry that connects the tangent and secant ratios of an angle A. It is derived directly from the Pythagorean theorem and is true for all angles where the tangent and secant are defined.
Simple Example
Quick Example
Imagine you are building a ramp for a toy car. If you know the angle of the ramp (A), and you can calculate the 'steepness' (tan A) and the 'length of the ramp relative to the base' (sec A), this identity tells you how these two values are always connected. It's like knowing that if you have a certain number of samosas (tan A), adding one more (1) gives you a related total (sec A) in a special mathematical way.
Worked Example
Step-by-Step
Let's verify the identity for A = 30 degrees.
Step 1: Find tan(30 degrees).
tan(30 degrees) = 1/sqrt(3)
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Step 2: Calculate tan^2(30 degrees).
tan^2(30 degrees) = (1/sqrt(3))^2 = 1/3
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Step 3: Calculate the Left Hand Side (LHS) of the identity.
LHS = 1 + tan^2(30 degrees) = 1 + 1/3 = 4/3
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Step 4: Find sec(30 degrees).
We know cos(30 degrees) = sqrt(3)/2.
sec(30 degrees) = 1/cos(30 degrees) = 1 / (sqrt(3)/2) = 2/sqrt(3)
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Step 5: Calculate the Right Hand Side (RHS) of the identity.
RHS = sec^2(30 degrees) = (2/sqrt(3))^2 = 4/3
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Step 6: Compare LHS and RHS.
Since LHS = 4/3 and RHS = 4/3, LHS = RHS.
Answer: The identity 1 + tan^2 A = sec^2 A is verified for A = 30 degrees.
Why It Matters
This identity helps simplify complex trigonometric expressions and solve problems in physics, engineering, and computer graphics. Engineers use it to design structures, while physicists apply it in wave mechanics and optics. Understanding this helps in careers like civil engineering, software development, and even satellite trajectory calculations at ISRO.
Common Mistakes
MISTAKE: Forgetting the '1' and writing tan^2 A = sec^2 A | CORRECTION: Always remember the '1' on the Left Hand Side. The identity is 1 + tan^2 A = sec^2 A.
MISTAKE: Confusing the identity with sin^2 A + cos^2 A = 1. | CORRECTION: While both are Pythagorean identities, they are distinct. This identity relates tangent and secant, not sine and cosine.
MISTAKE: Assuming tan^2 A is the same as tan(A^2). | CORRECTION: tan^2 A means (tan A) * (tan A). It is not the tangent of the angle A squared.
Practice Questions
Try It Yourself
QUESTION: If tan A = 3/4, find sec A using the identity 1 + tan^2 A = sec^2 A. | ANSWER: sec A = 5/4
QUESTION: Simplify the expression (sec^2 A - 1) / tan^2 A. | ANSWER: 1
QUESTION: Prove that (1 - cos^2 A) / (1 - sin^2 A) = tan^2 A. (Hint: Use basic identities first, then 1 + tan^2 A = sec^2 A) | ANSWER: (1 - cos^2 A) / (1 - sin^2 A) = sin^2 A / cos^2 A = (sin A / cos A)^2 = tan^2 A. We know 1 + tan^2 A = sec^2 A, so this is consistent.
MCQ
Quick Quiz
Which of the following is equivalent to sec^2 A - 1?
sin^2 A
cos^2 A
tan^2 A
cot^2 A
The Correct Answer Is:
C
From the identity 1 + tan^2 A = sec^2 A, if we subtract 1 from both sides, we get tan^2 A = sec^2 A - 1. Therefore, option C is correct.
Real World Connection
In the Real World
This identity is crucial in computer graphics and animation, especially when designing 3D models of buildings or characters. Game developers use trigonometry to calculate angles, distances, and object rotations, ensuring that virtual worlds look realistic. For instance, when designing a flying drone, understanding how angles affect its movement relies on these trigonometric relationships.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all possible values of the variables involved | TANGENT (tan A): Ratio of the opposite side to the adjacent side in a right-angled triangle | SECANT (sec A): Reciprocal of the cosine ratio, meaning 1/cos A | PYTHAGOREAN THEOREM: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
What's Next
What to Learn Next
Next, you should explore the third Pythagorean identity, 1 + cot^2 A = cosec^2 A. It's very similar to this one and builds on your understanding of how trigonometric ratios are connected. Mastering these identities will make solving more complex problems much easier!
