To derive the expression 1 + tan²x, we can start with a fundamental identity in trigonometry.
We know that:
1 + tan²x = sec²x
This identity comes from the Pythagorean identity involving sine and cosine:
sin²x + cos²x = 1
Now, if we divide each term by cos²x, we get:
tan²x = sin²x/cos²x
So, we can add 1 (which is cos²x/cos²x) to this expression:
1 + tan²x = cos²x/cos²x + sin²x/cos²x = (sin²x + cos²x)/cos²x = 1/cos²x = sec²x
Thus, the derived expression for 1 + tan²x is sec²x. This shows the deep relationship between the trigonometric functions.