Pythagorean Identity | (sec(x))^2 = (tan(x))^2 + 1

Pythagorean identity equal to sec^2(x)

The Pythagorean identity is a fundamental trigonometric identity that relates the three basic trigonometric functions: sine (sin), cosine (cos), and tangent (tan)

The Pythagorean identity is a fundamental trigonometric identity that relates the three basic trigonometric functions: sine (sin), cosine (cos), and tangent (tan). It states that for any angle x in a right triangle,

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

However, the question asks for the Pythagorean identity equal to sec^2(x). We can manipulate the original Pythagorean identity to obtain an equivalent identity using secant (sec) instead of cosine.

Starting with the original Pythagorean identity:

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

Dividing both sides by (cos(x))^2:

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

Using the reciprocal identity: (sin(x))^2 / (cos(x))^2 = (1 / (cos(x))^2) * (1 / (cos(x))^2)

After simplifying the right side:

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

Finally, we can rewrite the Pythagorean identity equal to (sec(x))^2 as:

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

This is the Pythagorean identity for secant squared. It shows that the square of the secant function is equal to the square of the tangent function plus 1.

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