Mastering The Pythagorean Trig Identity With Tangent: Derivation And Application

Pythagorean Trig Identity with tan

tan^2(x) + 1 = sec^2(x)

The Pythagorean trigonometric identity is a fundamental equation that relates trigonometric functions of any angle in a right triangle. It is given by:

sin²θ + cos²θ = 1

where θ is the angle in question, sin θ is the sine of θ, and cos θ is the cosine of θ.

Using this identity, we can derive the Pythagorean trig identity that involves the tangent function (tan θ) as follows:

tan²θ + 1 = sec²θ

where sec θ is the secant of θ.

To derive this identity, we first recall that the tangent function is defined as the ratio of the opposite and adjacent sides of a right triangle:

tan θ = opposite/adjacent

We can then use the Pythagorean theorem to relate the opposite and adjacent sides to the hypotenuse (the side opposite the right angle) as follows:

opposite² + adjacent² = hypotenuse²

Dividing both sides by adjacent², we get:

(opposite/adjacent)² + 1 = (hypotenuse/adjacent)²

Since the hypotenuse/adjacent ratio is the definition of the secant function, we can substitute sec θ for hypotenuse/adjacent and tan θ for opposite/adjacent to obtain:

tan²θ + 1 = sec²θ

And that’s our Pythagorean trig identity with tangent!

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