What is the Derivation of 1 + tan²A = sec²A?

S6-SA2-0220

Grade Level:

Class 10

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

Definition

What is it?

The derivation of 1 + tan²A = sec²A shows how this important trigonometric identity is proven using basic definitions of sine, cosine, and tangent, along with the Pythagorean identity. It's a fundamental relationship in trigonometry that connects tangent and secant functions for any angle A.

Simple Example

Quick Example

Imagine you are building a ramp for a toy car. If you know the angle of the ramp (A) and how 'steep' it is (tan A), this identity helps you quickly find a related value called 'sec A' which tells you something about the length of the ramp itself. It's like having a shortcut formula to find one value if you know another.

Worked Example

Step-by-Step

Let's derive 1 + tan²A = sec²A step-by-step:

Step 1: Start with the most basic trigonometric identity you know: sin²A + cos²A = 1. This is like the foundation of our house.
---Step 2: We want to get 'tan' and 'sec' into the picture. Remember that tan A = sin A / cos A and sec A = 1 / cos A.
---Step 3: To introduce 'tan A' and 'sec A' into our foundation equation (sin²A + cos²A = 1), let's divide every term in the equation by cos²A. This is allowed as long as cos A is not zero.
---Step 4: Dividing each term: (sin²A / cos²A) + (cos²A / cos²A) = (1 / cos²A).
---Step 5: Now, simplify each part. (sin²A / cos²A) is the same as (sin A / cos A)² which is tan²A.
---Step 6: (cos²A / cos²A) is simply 1.
---Step 7: (1 / cos²A) is the same as (1 / cos A)² which is sec²A.
---Step 8: Putting it all together, we get: tan²A + 1 = sec²A. Rearranging, we have 1 + tan²A = sec²A. This is our derived identity!

Why It Matters

This identity is super useful in fields like engineering, physics, and even AI/ML for calculating angles and distances precisely. Engineers use it to design structures, physicists to understand wave motion, and computer graphics artists to create realistic 3D models for games and movies.

Common Mistakes

MISTAKE: Forgetting to square the tan A and sec A terms, writing 1 + tan A = sec A. | CORRECTION: Always remember the squares! The identity is 1 + tan²A = sec²A, not 1 + tan A = sec A.

MISTAKE: Confusing the identities, for example, thinking it's 1 - tan²A = sec²A or 1 + sec²A = tan²A. | CORRECTION: The correct identity is 1 + tan²A = sec²A. It's helpful to derive it once or twice to remember the correct form.

MISTAKE: Dividing by sin²A instead of cos²A when starting from sin²A + cos²A = 1, leading to a different identity. | CORRECTION: To get tan²A and sec²A, you must divide by cos²A. Dividing by sin²A gives cot²A + 1 = cosec²A.

Practice Questions

Try It Yourself

QUESTION: If tan A = 3/4, find the value of sec A using the identity 1 + tan²A = sec²A. | ANSWER: sec A = 5/4

QUESTION: Prove that (sec A - tan A)(sec A + tan A) = 1 using the identity 1 + tan²A = sec²A. | ANSWER: (sec A - tan A)(sec A + tan A) = sec²A - tan²A. Since 1 + tan²A = sec²A, then sec²A - tan²A = 1. Hence proven.

QUESTION: If 1 + tan²A = sec²A, and sec A = 2, what is the value of tan A? (Assume A is an acute angle). | ANSWER: tan A = sqrt(3)

MCQ

Quick Quiz

Which basic identity is used as the starting point to derive 1 + tan²A = sec²A?

sin A + cos A = 1

sin²A + cos²A = 1

tan A = sin A / cos A

sec A = 1 / cos A

The Correct Answer Is:

The derivation starts by dividing the Pythagorean identity sin²A + cos²A = 1 by cos²A. Options C and D are definitions used in the steps, not the starting identity.

Real World Connection

In the Real World

Imagine ISRO scientists calculating the trajectory of a rocket or a satellite. They use trigonometry extensively. Identities like 1 + tan²A = sec²A help simplify complex calculations involving angles, velocities, and distances, ensuring the rocket reaches its destination accurately. It's also used in creating accurate maps and GPS systems for your mobile phone.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all values of the variables for which both sides are defined. | TANGENT (tan): In a right-angled triangle, the ratio of the length of the opposite side to the length of the adjacent side. | SECANT (sec): In a right-angled triangle, the ratio of the length of the hypotenuse to the length of the adjacent side (which is also 1/cos A). | PYTHAGOREAN IDENTITY: The fundamental trigonometric identity sin²A + cos²A = 1.

What's Next

What to Learn Next

Now that you understand 1 + tan²A = sec²A, you should explore the third Pythagorean identity, 1 + cot²A = cosec²A. It's derived similarly and will complete your understanding of these crucial trigonometric relationships, opening doors to solving more complex problems!