What is the exact value of csc^2 210degrees - cot^2 210degrees? Can you make a generalization about the value of csc^2 theta - cot^2 theta?
I'm not sure what to do, I'm familiar with the terms but not sure how to go about solving the question.
To find the value of csc^2 210 degrees - cot^2 210 degrees, let's start by understanding the trigonometric relationships involved.
The trigonometric functions csc (cosecant) and cot (cotangent) are related to sine and cosine as follows:
csc(theta) = 1/sin(theta)
cot(theta) = 1/tan(theta) = cos(theta)/sin(theta)
Now, let's evaluate csc^2 210 degrees:
We know that csc(theta) = 1/sin(theta), so csc^2(theta) = (1/sin(theta))^2 = 1/(sin^2(theta))
Since sin(210 degrees) is equal to sin(30 degrees) and sin(30 degrees) is equal to 1/2, we have:
csc^2 210 degrees = 1/(sin^2 210 degrees) = 1/(1/2)^2 = 4
Now, let's evaluate cot^2 210 degrees:
Since cot(theta) = cos(theta)/sin(theta), we can write cot^2(theta) as (cos(theta)/sin(theta))^2 = cos^2(theta)/sin^2(theta).
Since cos(210 degrees) is equal to -cos(30 degrees) and cos(30 degrees) is equal to √3/2, we have:
cot^2 210 degrees = cos^2 210 degrees / sin^2 210 degrees = (√3/2)^2 / (1/2)^2 = (3/4) / (1/4) = 3
Finally, subtracting cot^2 210 degrees from csc^2 210 degrees:
csc^2 210 degrees - cot^2 210 degrees = 4 - 3 = 1.
Therefore, the exact value of csc^2 210 degrees - cot^2 210 degrees is 1.
Now let's generalize the value of csc^2(theta) - cot^2(theta) for any angle theta:
Using the relationships above, we can rewrite csc^2(theta) - cot^2(theta) as 1/sin^2(theta) - cos^2(theta)/sin^2(theta).
Combining the fractions, we have:
csc^2(theta) - cot^2(theta) = (1 - cos^2(theta))/sin^2(theta).
Now, using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can substitute 1 - cos^2(theta) with sin^2(theta):
csc^2(theta) - cot^2(theta) = sin^2(theta)/sin^2(theta) = 1.
So, the generalization is that csc^2(theta) - cot^2(theta) equals 1 for any angle theta.