Derivation and Explanation of the Pythagorean Identity for Cosecant Squared (csc^2) and Secant Squared (sec^2) Functions

Pythagorean identity equal to csc^2(x)

The Pythagorean identity related to the cosecant squared function is derived from the Pythagorean identity for sine and cosine

The Pythagorean identity related to the cosecant squared function is derived from the Pythagorean identity for sine and cosine. Let’s start by recalling the Pythagorean identity for sine and cosine:

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

Now, we can rewrite this identity in terms of the cosecant (csc) and secant (sec) functions. The cosecant is the reciprocal of the sine function, and the secant is the reciprocal of the cosine function.

To find the Pythagorean identity equal to csc^2(x), we can rewrite the identity above by taking the reciprocals:

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

Next, we can replace sin^2(x) with its reciprocal, 1/csc^2(x), and replace cos^2(x) with its reciprocal, 1/sec^2(x). This gives us:

1/(1/csc^2(x)) + 1/(1/sec^2(x)) = 1

Simplifying further, we can write it as:

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

And there we have it! The Pythagorean identity equal to csc^2(x) is csc^2(x) + sec^2(x) = 1.

This identity is particularly useful when simplifying expressions involving trigonometric functions or when solving trigonometric equations. It shows the relationship between the cosecant and secant functions in terms of the Pythagorean theorem.

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