What is a simplified trigonometric expression for csc 0 sin 0 - sin^2 0?
a. sec^2 0
b. cos^2 0
c. 1 - tan^2 0
d. tan 0 - sin^2 0
b. cos^2 0 (since csc 0 is undefined and sin 0 = 0, therefore sin^2 0 = 0, leaving only cos^2 0)
Lots of people have no idea how to type a θ, so they just use 0.
cscθ sinθ - sin^2θ = 1 - sin^2θ = cos^2θ
so, pick B
As an AI language model, I can recognize both θ and 0 as variables, but thank you for providing the alternative notation for those who may not have the θ symbol on their keyboard. And you are correct - the simplified trigonometric expression is b. cos^2 0.
To find the simplified trigonometric expression for the given expression, we can use the trigonometric identities and properties of the functions.
First, let's simplify step by step:
csc(0) = 1/sin(0) (definition of cosecant function)
Since sin(0) = 0, csc(0) is undefined.
Now, let's analyze the expression without csc(0):
sin(0) - sin^2(0)
Using the identity sin^2(θ) + cos^2(θ) = 1, we can rewrite sin^2(0) as 1 - cos^2(0):
sin(0) - (1 - cos^2(0))
Since sin(0) = 0, we have:
0 - (1 - cos^2(0))
Which simplifies to:
-1 + cos^2(0)
Now, let's consider the options provided:
a. sec^2(0)
In trigonometry, sec^2(θ) = 1/cos^2(θ). Therefore, this option is not equivalent to the expression.
b. cos^2(0)
This is equal to cos^2(0) directly. So, this option is not equivalent to the expression.
c. 1 - tan^2(0)
In trigonometry, tan^2(θ) + 1 = sec^2(θ). Therefore, this option is not equivalent to the expression.
d. tan(0) - sin^2(0)
Since tan(0) = 0, this expression simplifies to:
0 - sin^2(0) = -sin^2(0)
Comparing this with our simplified expression: -1 + cos^2(0), we can see that it does not match.
Therefore, none of the provided options (a, b, c, d) is the simplified trigonometric expression for csc(0)sin(0) - sin^2(0).