Understanding the Pythagorean Identity: The Connection between Tangent and Secant Functions

Pythagorean Identity: tan^2(x) + 1 =

The Pythagorean Identity related to the tangent function is as follows:

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

Let’s break down this identity and understand how it is derived

The Pythagorean Identity related to the tangent function is as follows:

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

Let’s break down this identity and understand how it is derived.

To begin with, let’s consider a right triangle with one acute angle x. In this triangle, the tangent of x is defined as the ratio of the length of the side opposite x (denoted by a) to the length of the adjacent side (denoted by b). So, tan(x) = a/b.

Now, let’s express the tangent in terms of the sine and cosine functions:

tan(x) = sin(x)/cos(x)

Squaring both sides:

tan^2(x) = (sin(x)/cos(x))^2 = sin^2(x)/cos^2(x)

Now, let’s express the sine and cosine functions in terms of their reciprocal functions:

sin^2(x) = (1 – cos^2(x)) and cos^2(x) = (1 – sin^2(x))

Substituting these values into the expression for tan^2(x), we get:

tan^2(x) = (1 – cos^2(x))/(1 – sin^2(x))

Now, using the reciprocal identities:

sec^2(x) = 1/cos^2(x) and csc^2(x) = 1/sin^2(x)

We can rewrite the expression for tan^2(x) as:

tan^2(x) = (1 – cos^2(x))/(1 – sin^2(x)) = 1/cos^2(x) = sec^2(x)

So, the Pythagorean Identity for tangent is tan^2(x) + 1 = sec^2(x).

This identity is useful in various trigonometric calculations and proofs. It relates the squared tangent function to the squared secant function, highlighting their connection in right triangles.

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