17. The expression

2
tan
𝜗
1−
tan2
𝜗
is identically equal to
A. cos 2𝜃
B. sin 2𝜃
C. tan 2𝜃
D. cot 2�

1 answer

We can start by simplifying the expression using the trigonometric identity:

tan 2𝜃 = 2 tan 𝜃 / (1 - tan^2 𝜃)

Letting x = tan 𝜃, we can rewrite the expression as:

2x / (1 - x^2)

To get a common denominator, we can rewrite 1 as (1 - x^2 + x^2):

2x / ((1 - x^2) + x^2)

Simplifying, we get:

2x / (cos^2 𝜃)

Using the identity cos^2 𝜃 = 1 / (1 + tan^2 𝜃), we can rewrite this as:

2x(1 + x^2)

Expanding and simplifying, we get:

2x + 2x^3

Substituting back in tan 𝜃 for x, we get:

2 tan 𝜃 + 2 tan^3 𝜃

This is equivalent to:

2 tan 𝜃 (1 + tan^2 𝜃)

Using the identity 1 + tan^2 𝜃 = sec^2 𝜃, we can rewrite this as:

2 tan 𝜃 sec^2 𝜃

Finally, using the identity sec^2 𝜃 = 1 + tan^2 𝜃, we get:

2 tan 𝜃 (1 + tan^2 𝜃) / (1 + tan^2 𝜃)

Simplifying, we get:

2 tan 𝜃

Therefore, the expression is identically equal to B. sin 2𝜃.