Understanding the Pythagorean Identity | The Relationship between Cosine, Tangent, and the Value 1

Pythagorean Identity1 + tan² θ

The Pythagorean Identity is a fundamental trigonometric identity that relates the values of sine (sin), cosine (cos), and tangent (tan) of an angle in a right triangle

The Pythagorean Identity is a fundamental trigonometric identity that relates the values of sine (sin), cosine (cos), and tangent (tan) of an angle in a right triangle. The Pythagorean Identity states that for any angle θ in a right triangle:

sin² θ + cos² θ = 1

However, the expression you provided, 1 + tan² θ, is another form of the Pythagorean Identity that involves the tangent function. To understand this, let’s rewrite the original identity in terms of tan:

1 = sin² θ + cos² θ
1 = (sin θ / cos θ)² + cos² θ
1 = (sin² θ / cos² θ) + cos² θ

Now, since sin θ / cos θ = tan θ, we can substitute this into the equation:

1 = (tan θ)² + cos² θ

Finally, by rearranging the terms, we get:

1 + tan² θ = cos² θ + (tan θ)²

So, 1 + tan² θ is just an alternative form of the Pythagorean Identity, which highlights the relationship between the tangent function, cosine squared, and the value 1.

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