Proving the Trigonometric Identity sec2A - tan2A 1

Many students encounter confusion when dealing with the various trigonometric identities, especially the one that often gets misinterpreted as:

Which Equation is Correct?

The equation that is frequently mistaken is:

sec2A / cos2A - 1 / cos2A

This simplifies to:

sec2A - 1 / cos2A

And further reduces to:

-1 / cos2A

However, the correct identity is:

sec2A - tan2A 1

This is a fundamental trigonometric identity that is often used in solving various trigonometric equations and simplifying expressions.

Proof of sec2A - tan2A 1

The correct identity, sec2A - tan2A 1, can be proven using basic trigonometric definitions and the Pythagorean identity. Let's break it down step-by-step:

Using Basic Definitions

The definitions of secθ and tanθ are:

tanθ sinθ / cosθ

secθ 1 / cosθ

Substitution and Simplification

Substitute these definitions into the identity:

sec2θ - tan2θ 1

This becomes:

(1 / cos2θ) - (sin2θ / cos2θ) 1

Distribute and combine the fractions:

(1 - sin2θ) / cos2θ 1

Recall the fundamental Pythagorean identity:

cos2θ sin2θ 1

From this, we can express 1 - sin2θ as cos2θ:

(cos2θ / cos2θ) 1

This simplifies to:

1 1

This proves that the identity sec2θ - tan2θ 1 is indeed correct and valid in all cases.

Axiom and Fundamental Identity

The Pythagorean identity, cos2θ sin2θ 1, is a fundamental axiom in trigonometry. It can be derived from the unit circle, where the radius R1, and the Pythagorean theorem applied to a right triangle:

x2 y2 R2

By substituting R1:

x2 y2 1

Here, x and y represent the cosine and sine values, respectively. This means that for any angle θ:

cos2θ sin2θ 1

Even in cases where x or y is negative, the squared values will still sum to 1, ensuring the validity of the identity.

Conclusion

Therefore, the identity sec2θ - tan2θ 1, along with its fundamental underpinning, the Pythagorean identity, are essential in understanding and working with trigonometric functions and equations. Misunderstandings about these identities can often lead to errors in calculations and problem-solving.

Note: The incorrect equation given in the introduction is a common mistake that arises from a misunderstanding of the trigonometric identities. By carefully applying the definitions and the Pythagorean identity, we can resolve such confusion and ensure accurate results in trigonometric computations.