To simplify the expression 1 + tan²(x), you can use a fundamental identity from trigonometry known as the Pythagorean identity. According to this identity:
1 + tan²(x) = sec²(x)
Here’s a breakdown of the steps involved in this simplification:
- Recall the definition of tan(x):
- tan(x) is defined as the ratio of the opposite side to the adjacent side in a right triangle, or tan(x) = sin(x)/cos(x).
- Using the definition, we can express tan²(x):
- tan²(x) = (sin²(x)/cos²(x)).
- Now, substitute this into the original expression:
- 1 + tan²(x) = 1 + (sin²(x)/cos²(x)).
- To combine the terms, get a common denominator:
- 1 can be rewritten as (cos²(x)/cos²(x)), so:
- 1 + tan²(x) = (cos²(x)/cos²(x)) + (sin²(x)/cos²(x)) = (cos²(x) + sin²(x))/cos²(x).
- According to the Pythagorean identity, we know that:
- sin²(x) + cos²(x) = 1.
- Thus, we can substitute this identity back into our expression:
- (1)/cos²(x) = sec²(x).
In conclusion, the simplified form of the expression 1 + tan²(x) is sec²(x). This identity is very useful in various applications within calculus and trigonometry, particularly in integrals and derivatives involving trigonometric functions.