What does tan squared plus 1 equal according to trigonometric identities?

The trigonometric identity states that ( \tan^2(\theta) + 1 = \sec^2(\theta) ). This identity is fundamental in trigonometry and can be derived from the definitions of the tangent and secant functions.

To see why this holds true, consider the definitions of tangent and secant in terms of sine and cosine:

• ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )

• ( \sec(\theta) = \frac{1}{\cos(\theta)} )

If we square the tangent function:

[

\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}

]

Adding 1 to both sides gives:

[

\tan^2(\theta) + 1 = \frac{\sin^2(\theta)}{\cos^2(\theta)} + 1

]

To combine the terms, we express 1 as (\frac{\cos^2(\theta)}{\cos^2(\theta)}):

[

\tan^2(\theta) + 1 = \frac{\sin^2(\theta)}{\cos^2(\