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What is the Identity 1 + tan²A = sec²A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity 1 + tan²A = sec²A is a fundamental trigonometric relationship that connects the tangent and secant ratios of an angle A. It is derived from the Pythagorean identity sin²A + cos²A = 1 and is true for all angles where tan A and sec A 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 the 'steepness' (tan A), this identity helps you quickly find a related value called 'sec A'. It's like having a special calculator button that gives you one value if you know the other, without needing to know the actual sides of the triangle.
Worked Example
Step-by-Step
Let's verify the identity for A = 30 degrees.
Step 1: Find tan(30 degrees). We know tan(30 degrees) = 1/sqrt(3).
---Step 2: Calculate tan²(30 degrees). tan²(30 degrees) = (1/sqrt(3))² = 1/3.
---Step 3: Calculate 1 + tan²(30 degrees). 1 + 1/3 = 4/3.
---Step 4: Find sec(30 degrees). We know cos(30 degrees) = sqrt(3)/2, so sec(30 degrees) = 1/cos(30 degrees) = 2/sqrt(3).
---Step 5: Calculate sec²(30 degrees). sec²(30 degrees) = (2/sqrt(3))² = 4/3.
---Step 6: Compare the results. Since 1 + tan²(30 degrees) = 4/3 and sec²(30 degrees) = 4/3, the identity holds true for A = 30 degrees.
Answer: The identity 1 + tan²A = sec²A is verified for A = 30 degrees.
Why It Matters
This identity is crucial in fields like Physics for analyzing wave motion and in Engineering for designing structures. It helps scientists and engineers simplify complex equations, making calculations easier and more accurate. Understanding it can open doors to careers in space technology or even AI/ML.
Common Mistakes
MISTAKE: Writing 1 + tan A = sec A | CORRECTION: Remember the squares! It's 1 + tan²A = sec²A, not without the squares.
MISTAKE: Using the identity for angles where tan A or sec A are undefined (e.g., A = 90 degrees) | CORRECTION: Always ensure the trigonometric ratios involved are defined for the given angle. tan 90 degrees and sec 90 degrees are undefined.
MISTAKE: Confusing it with 1 + cot²A = cosec²A | CORRECTION: Keep track of which ratios go together. tan and sec are linked in this identity, while cot and cosec are linked in another.
Practice Questions
Try It Yourself
QUESTION: If tan A = 3/4, find the value of sec²A. | ANSWER: sec²A = 1 + tan²A = 1 + (3/4)² = 1 + 9/16 = 25/16.
QUESTION: If sec A = 5/3, find the value of tan²A. | ANSWER: tan²A = sec²A - 1 = (5/3)² - 1 = 25/9 - 1 = 25/9 - 9/9 = 16/9.
QUESTION: Simplify (sec²A - tan²A) / (1 + tan²A). | ANSWER: We know sec²A - tan²A = 1 and 1 + tan²A = sec²A. So, the expression becomes 1 / sec²A = cos²A.
MCQ
Quick Quiz
Which of the following is equivalent to sec²A - 1?
sin²A
cos²A
tan²A
cot²A
The Correct Answer Is:
C
From the identity 1 + tan²A = sec²A, we can rearrange it to get tan²A = sec²A - 1. So, option C is correct.
Real World Connection
In the Real World
This identity is used by civil engineers in India to calculate the stability of structures like bridges and buildings, especially when dealing with angles and forces. For example, when designing a flyover in Mumbai, understanding how different angles affect the strength of support beams involves trigonometry, including identities like this one.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all possible values of the variables | TANGENT (tan): Ratio of opposite side to adjacent side in a right-angled triangle | SECANT (sec): Reciprocal of the cosine ratio (1/cos A) | TRIGONOMETRY: Branch of mathematics dealing with relationships between sides and angles of triangles | PYTHAGOREAN IDENTITY: Fundamental trigonometric identity derived from the Pythagorean theorem (sin²A + cos²A = 1)
What's Next
What to Learn Next
Great job understanding this identity! Next, you should explore the identity 1 + cot²A = cosec²A. It's very similar and builds directly on your understanding of this one, helping you master all the core trigonometric identities.
