17. The expression
2
tan
π
1β
tan2
π
is identically equal to
A. cos 2π
B. sin 2π
C. tan 2π
D. cot 2οΏ½
We can start by simplifying the expression using the trigonometric identity:
tan 2π = 2 tan π / (1 - tan^2 π)
Letting x = tan π, 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 π)
Using the identity cos^2 π = 1 / (1 + tan^2 π), we can rewrite this as:
2x(1 + x^2)
Expanding and simplifying, we get:
2x + 2x^3
Substituting back in tan π for x, we get:
2 tan π + 2 tan^3 π
This is equivalent to:
2 tan π (1 + tan^2 π)
Using the identity 1 + tan^2 π = sec^2 π, we can rewrite this as:
2 tan π sec^2 π
Finally, using the identity sec^2 π = 1 + tan^2 π, we get:
2 tan π (1 + tan^2 π) / (1 + tan^2 π)
Simplifying, we get:
2 tan π
Therefore, the expression is identically equal to B. sin 2π.
To simplify the expression, we will use the trigonometric identity:
tan^2(π) + 1 = sec^2(π)
Let's rewrite the expression:
2tan(π) / (1 - tan^2(π))
Since we have a tan^2(π) in the denominator, we can rewrite it as sec^2(π):
2tan(π) / (1 - sec^2(π))
Now, we can use the identity:
1 - sec^2(π) = -cos^2(π)
Substituting this identity into the expression:
2tan(π) / (-cos^2(π))
Next, since tan(π) = sin(π) / cos(π), we can rewrite the expression as:
2(sin(π) / cos(π)) / (-cos^2(π))
Simplifying further:
-2sin(π) / cos(π) * 1 / cos(π)
-2sin(π) / cos^2(π)
Finally, since sin(π) / cos(π) is equal to tan(π), we have:
-2tan(π)
Therefore, the expression is equal to -2tan(π).
So, the answer is not provided in the given options.