How can I simplify the expression 1 + tan^2(x)?

To simplify the expression 1 + tan²(x), we can utilize a well-known identity from trigonometry.

The identity states:

• 1 + tan²(x) = sec²(x)

Here’s the step-by-step explanation:

1. Recall that the tangent function is defined as:

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

2. Using this definition, we know:

   tan²(x) = (sin²(x) / cos²(x))

3. Substituting this into the expression:

   1 + tan²(x) = 1 + (sin²(x) / cos²(x))

4. To combine these fractions, note that:

   1 = cos²(x) / cos²(x)

5. Now we can rewrite the entire expression:

   1 + tan²(x) = (cos²(x) + sin²(x)) / cos²(x)

6. By the Pythagorean identity, we know that:

   sin²(x) + cos²(x) = 1

7. Thus, we can simplify this further:

   (1) / cos²(x) = sec²(x)

In conclusion, the simplified form of 1 + tan²(x) is:

sec²(x)

This identity is significant in trigonometry and calculus, particularly when it comes to integration and differentiation involving trigonometric functions.

Understanding this simplification can help you tackle more complex trigonometric problems with confidence!