What is the Derivation of 1 + tan^2 A = sec^2 A?

S6-SA2-0059

Grade Level:

Class 10

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

Definition

What is it?

The derivation of 1 + tan^2 A = sec^2 A shows how this important trigonometric identity comes directly from the basic Pythagorean identity (sin^2 A + cos^2 A = 1) by dividing all its terms by cos^2 A. This identity helps simplify complex trigonometric expressions and solve problems involving angles and triangles.

Simple Example

Quick Example

Imagine you have a cricket field where the pitch length (adjacent side) is 30 meters and the distance a fielder throws the ball to the boundary (opposite side) is 40 meters. If you know the angle 'A' the throw makes with the pitch, this identity helps relate the tangent of that angle (opposite/adjacent) to the secant of that angle (hypotenuse/adjacent) in a direct way.

Worked Example

Step-by-Step

Let's derive 1 + tan^2 A = sec^2 A from sin^2 A + cos^2 A = 1.
Step 1: Start with the fundamental trigonometric identity: sin^2 A + cos^2 A = 1.
---Step 2: To get '1' on the left side and 'tan^2 A' and 'sec^2 A' terms, we need to divide the entire equation by cos^2 A. (Remember, cos A cannot be zero).
---Step 3: Divide each term by cos^2 A: (sin^2 A / cos^2 A) + (cos^2 A / cos^2 A) = (1 / cos^2 A).
---Step 4: Simplify each term. We know that sin A / cos A = tan A, so sin^2 A / cos^2 A = tan^2 A.
---Step 5: Also, cos^2 A / cos^2 A = 1.
---Step 6: And we know that 1 / cos A = sec A, so 1 / cos^2 A = sec^2 A.
---Step 7: Substitute these simplified terms back into the equation: tan^2 A + 1 = sec^2 A.
---Step 8: Rearranging it gives us: 1 + tan^2 A = sec^2 A. This is the derived identity.

Why It Matters

Understanding this identity is crucial in fields like Engineering and Physics to calculate forces, trajectories, and wave patterns. It helps engineers design stable bridges and scientists predict satellite orbits. Even AI/ML algorithms use trigonometric functions for pattern recognition and signal processing.

Common Mistakes

MISTAKE: Students often confuse the identity with 1 + cot^2 A = cosec^2 A. | CORRECTION: Remember that 'tan' is related to 'sec', and 'cot' is related to 'cosec'. They are derived from dividing by cos^2 A and sin^2 A respectively.

MISTAKE: Forgetting to divide ALL terms by cos^2 A, leading to an incorrect identity. | CORRECTION: Always divide every single term (left and right side) of the original identity (sin^2 A + cos^2 A = 1) by cos^2 A.

MISTAKE: Assuming the identity holds even when cos A = 0 (i.e., for A = 90 degrees or 270 degrees). | CORRECTION: The derivation involves dividing by cos^2 A, so the identity is not defined for angles where cos A is zero, as division by zero is undefined.

Practice Questions

Try It Yourself

QUESTION: If tan A = 3/4, find sec^2 A using the identity. | ANSWER: sec^2 A = 1 + (3/4)^2 = 1 + 9/16 = 16/16 + 9/16 = 25/16

QUESTION: Prove that (sec A - tan A)(sec A + tan A) = 1 using the identity. | ANSWER: (sec A - tan A)(sec A + tan A) = sec^2 A - tan^2 A. From 1 + tan^2 A = sec^2 A, we know sec^2 A - tan^2 A = 1. Hence proved.

QUESTION: If sec^2 A = 5/4, find the value of tan^2 A. | ANSWER: Using 1 + tan^2 A = sec^2 A, we have 1 + tan^2 A = 5/4. So, tan^2 A = 5/4 - 1 = 5/4 - 4/4 = 1/4.

MCQ

Quick Quiz

Which basic identity is used to derive 1 + tan^2 A = sec^2 A?

sin^2 A - cos^2 A = 1

sin^2 A + cos^2 A = 1

1 - tan^2 A = sec^2 A

1 + cot^2 A = cosec^2 A

The Correct Answer Is:

The identity 1 + tan^2 A = sec^2 A is derived by dividing the fundamental identity sin^2 A + cos^2 A = 1 by cos^2 A. Options A, C, and D are either incorrect or are different identities.

Real World Connection

In the Real World

This identity is used by ISRO scientists and engineers to calculate the precise angles for satellite dish antennas to receive signals from space. Understanding how angles relate to each other helps them point the dish accurately, ensuring clear communication with our satellites like Chandrayaan and Mangalyaan.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all valid values of the variables. | TANGENT (tan A): Ratio of the opposite side to the adjacent side in a right-angled triangle. | SECANT (sec A): Ratio of the hypotenuse to the adjacent side in a right-angled triangle (which is 1/cos A). | DERIVATION: The process of showing how a formula or statement is obtained from other basic principles.

What's Next

What to Learn Next

Next, you should learn about the derivation of 1 + cot^2 A = cosec^2 A. It's very similar to this one, but you'll divide by a different term! Mastering these derivations will make you super confident in solving any trigonometry problem.