Understanding the Pythagorean Identity: Expressing it in terms of sec^2(x)

Pythagorean identity equal to sec^2(x)

tan^2(x) + 1

The Pythagorean identity states that for any angle x in a right triangle,

sin^2(x) + cos^2(x) = 1.

To express the Pythagorean identity in terms of sec^2(x), we can manipulate the equation using definitions of trigonometric functions.

Starting with the Pythagorean identity:

sin^2(x) + cos^2(x) = 1,

Let’s divide both sides of the equation by cos^2(x):

(sin^2(x) + cos^2(x)) / cos^2(x) = 1 / cos^2(x).

Using the fact that (1 / cos^2(x)) is equivalent to sec^2(x), we can rewrite the equation as:

sin^2(x) / cos^2(x) + cos^2(x) / cos^2(x) = sec^2(x).

Now, let’s simplify the equation:

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

Therefore, the Pythagorean identity expressed in terms of sec^2(x) is:

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

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