Question

Simplify the following: tan 22.5° / (1 - tan² 22.5°)

a) √2 - 1
b) √2 + 1
c) 2 - √2
d) 2 + √2

Answer 1

To simplify the expression tan 22.5° / (1 - tan² 22.5°), we can use the identity for the tangent squared and the Pythagorean identity to simplify the expression. Substituting the values of sin 22.5° and cos 22.5°, the simplified expression is √2 - 1.

Step-by-step explanation:

To simplify the expression tan 22.5° / (1 - tan² 22.5°), we can use the identity for the tangent squared: tan² θ = (sin θ)² / (cos θ)². Substituting this into the expression, we get:

tan 22.5° / (1 - (sin 22.5° / cos 22.5°)²)

Simplifying further using the Pythagorean identity sin² θ + cos² θ = 1, we have:

tan 22.5° / (1 - sin² 22.5° / cos² 22.5°)

Now, we can substitute the values of sin 22.5° = √2 - 1 and cos 22.5° = √2 + 1:

(√2 - 1) / (1 - (√2 - 1)² / (√2 + 1)²)

Simplifying the expression, we get √2 - 1, so the answer is a) √2 - 1.