What is secant squared equal to?

Understand the Problem

The question is asking for the equivalent expression or value for secant squared in trigonometry, particularly in terms of sine or cosine functions. This revolves around the identity related to secant, which is the reciprocal of cosine.

Answer

$$ \sec^2(x) = \frac{1}{1 - \sin^2(x)} $$

Steps to Solve

Recall the Secant Definition

The secant function is defined as the reciprocal of the cosine function. Thus, we have:

$$ \sec(x) = \frac{1}{\cos(x)} $$

Square the Secant Function

Next, we square the secant function:

$$ \sec^2(x) = \left( \frac{1}{\cos(x)} \right)^2 $$

This results in:

$$ \sec^2(x) = \frac{1}{\cos^2(x)} $$

Use the Pythagorean Identity

In trigonometry, we use the Pythagorean identity:

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

From this identity, we can express $\sin^2(x)$ in terms of $\cos^2(x)$:

$$ \sin^2(x) = 1 - \cos^2(x) $$

Substitute into the Secant Squared Expression

We can now relate $\sec^2(x)$ with sine using the reciprocal identity. From the squared secant definition, we have:

To express in terms of sine, we rearrange $\sec^2(x)$:

$$ \sec^2(x) = \frac{1}{\cos^2(x)} = \frac{1}{1 - \sin^2(x)} $$

This gives us the equivalent expression.

The equivalent expression for secant squared in terms of sine is:

$$ \sec^2(x) = \frac{1}{1 - \sin^2(x)} $$

More Information

Secant squared is crucial in trigonometry, especially when dealing with derivatives in calculus or solving integrals. This relationship to sine also highlights the connection between different trigonometric functions and their identities.

Tips

Confusing Identities: A common mistake is mixing up the reciprocal identities; remember that $\sec(x)$ is always the reciprocal of $\cos(x)$.
Neglecting Pythagorean Identities: It's easy to forget the relationship given by the Pythagorean identity, which is essential when rearranging expressions.

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