Understanding the Pythagorean Trig Identity | Deriving the Relationship between Tangent and √2

Pythagorean Trig Identity with tan

The Pythagorean trigonometric identity involving the tangent (tan) function is derived from the Pythagorean theorem

The Pythagorean trigonometric identity involving the tangent (tan) function is derived from the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The tangent function is defined as the ratio of the lengths of the opposite side and the adjacent side of a right triangle. We can derive the Pythagorean trig identity involving the tangent function as follows:

Consider a right triangle with sides of lengths a, b, and c, where c is the hypotenuse. The tangent of an angle θ in this triangle can be defined as:
tan(θ) = a/b

Squaring both sides of the equation gives us:
tan^2(θ) = (a/b)^2

Since a^2/b^2 is equal to (a/b)^2, we can rewrite the equation as:
tan^2(θ) = a^2/b^2

Now, using the Pythagorean theorem, we know that a^2 + b^2 = c^2. Substituting this into the equation, we get:
tan^2(θ) = (a^2 + b^2)/b^2

Finally, simplifying the equation, we have:
tan^2(θ) = 1 + (b^2/b^2)

Simplifying further:
tan^2(θ) = 1 + 1

tan^2(θ) = 2

Taking the square root of both sides, we get the Pythagorean trig identity involving the tangent:
tan(θ) = √2

So, the Pythagorean trig identity involving the tangent function is tan(θ) = √2.

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