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Pythagorean Identities in Mathematics

The Most Powerful Equations in Trigonometry

Of all the tools in trigonometry, the Pythagorean Identities are the most essential. They are equations that relate the sine, cosine, and tangent functions to each other using the Pythagorean Theorem. They are fundamentally "true" for any angle θ.

These identities are the key to simplifying complex expressions in Calculus, Physics, and Engineering. If you see a "squared" trig function, chances are you will need a Pythagorean Identity.

1. The Primary Identity

[Image of Pythagorean Identity unit circle derivation]

This identity is derived directly from the Unit Circle. In a unit circle (radius $r=1$), any point on the circle has coordinates $(x, y)$ where $x = \cos(\theta)$ and $y = \sin(\theta)$.

Applying the Pythagorean Theorem ($a^2 + b^2 = c^2$) to the triangle inside the circle gives us:

\sin^2(\theta) + \cos^2(\theta) = 1

This means that for ANY angle, the square of the sine plus the square of the cosine will always equal 1. This allows us to rewrite formulas: $\sin^2(\theta) = 1 - \cos^2(\theta)$.

2. Deriving the Second Identity (Tangent & Secant)

We can create new identities by dividing the primary equation. If we divide every term in the first identity by $\cos^2(\theta)$:

$\frac{\sin^2(\theta)}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)}$

Knowing that $\tan = \sin/\cos$ and $\sec = 1/\cos$, this simplifies to:

\tan^2(\theta) + 1 = \sec^2(\theta)

3. Deriving the Third Identity (Cotangent & Cosecant)

Similarly, if we divide every term in the primary identity by $\sin^2(\theta)$:

$\frac{\sin^2(\theta)}{\sin^2(\theta)} + \frac{\cos^2(\theta)}{\sin^2(\theta)} = \frac{1}{\sin^2(\theta)}$

Knowing that $\cot = \cos/\sin$ and $\csc = 1/\sin$, this simplifies to:

1 + \cot^2(\theta) = \csc^2(\theta)

4. Summary of the Three Identities

Memorizing these three forms allows you to swap between functions effortlessly:

  • Sine/Cosine: $\sin^2\theta + \cos^2\theta = 1$
  • Tangent/Secant: $\tan^2\theta + 1 = \sec^2\theta$
  • Cotangent/Cosecant: $1 + \cot^2\theta = \csc^2\theta$

5. Example Problem

Simplify the expression: $(1 - \sin^2x) \cdot \sec^2x$

  1. Recognize that $1 - \sin^2x$ is a variation of the first identity. It equals $\cos^2x$.
  2. Substitute it in: $\cos^2x \cdot \sec^2x$
  3. Recall that $\sec x = 1/\cos x$, so $\sec^2x = 1/\cos^2x$.
  4. Multiply: $\cos^2x \cdot \frac{1}{\cos^2x} = 1$

The entire messy expression simplifies to just 1.

Conclusion

The Pythagorean Identities are the backbone of analytic trigonometry. By connecting the square of a function to the number 1, they provide a powerful bridge between geometry and algebra.