Final answer:
To simplify the expression (1 - tan²θ) / (1 + tan²θ), we use trigonometric identities to show that it is equivalent to cos(2θ).
Step-by-step explanation:
To simplify the trigonometric expression (1 - tan²θ) / (1 + tan²θ), we can use a well-known trigonometric identity involving tanθ. The identity states that 1 + tan²θ = sec²θ, where secθ is the secant of θ. Replacing the denominator of our original expression with this identity gives us:
(1 - tan²θ) / sec²θ
Since sec²θ is equivalent to 1/cos²θ, we can write:
(1 - tan²θ) * cos²θ
Now, using the identity sin²θ + cos²θ = 1, and the fact that tan²θ = sin²θ / cos²θ, our expression simplifies to:
(1 - sin²θ / cos²θ) * cos²θ
Further simplifying, we have:
cos²θ - sin²θ
Finally, we arrive at the trigonometric identity cos(2θ), which is equivalent to cos²θ - sin²θ. Therefore, the simplest form of the expression is:
cos(2θ)