The Pythagorean Identity: A Fundamental Trigonometric Concept For Right-Angled Triangles

pythagorean identity (tan and sec)

tan²θ + 1 = sec²θ

The Pythagorean identity for tangent and secant is a trigonometric equation that relates the values of tangent and secant of an angle in a right triangle. It states that:

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

where theta is the measure of the angle in radians or degrees.

This identity can be derived from the Pythagorean theorem, which relates the sides of a right triangle. In a right triangle with one angle of measure theta, the opposite side is of length a, the adjacent side is of length b and the hypotenuse is of length c. The Pythagorean theorem states that:

a^2 + b^2 = c^2

Dividing both sides of the equation by c^2, we get:

(a^2)/c^2 + (b^2)/c^2 = 1

Using the definitions of tangent and secant, we can write:

tan(theta) = a/b
sec(theta) = c/b

Squaring both sides of the equation for tangent and substituting the values of a and b, we get:

tan^2(theta) = (a/b)^2 = (c^2 – b^2)/b^2

Substituting the equation obtained from the Pythagorean theorem for c^2 – b^2, we get:

tan^2(theta) = [(a^2)/(c^2)]/[1 – (b^2)/(c^2)]

Substituting the equation for secant, we get:

tan^2(theta) = (1/sec^2(theta))/(1 – tan^2(theta))

Multiplying both sides of the equation by sec^2(theta)(1 – tan^2(theta)), we get:

tan^2(theta)sec^2(theta) + sec^2(theta) = tan^2(theta) + sec^4(theta)

Rearranging the terms, we get:

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

This is the Pythagorean identity for tangent and secant.

This identity is useful in simplifying trigonometric expressions and solving trigonometric equations. For example, if we have an expression involving both tangent and secant, we can use the Pythagorean identity to transform it into an expression involving only one of the two functions. Similarly, if we have an equation involving both tangent and secant, we can use the Pythagorean identity to simplify the equation and solve for the unknown variable.

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