for all values for which the following is defined, find the value of cot^2(theta)-csc^2(theta)
Substituting
cot(θ)=cos(θ)/sin(θ)
and csc(θ) = 1/sin(θ)
you will have a common denominator of
sin²(θ).
Add the numerators and use
cos²(θ)+sin²(θ) = 1
to simplify further.
so wwould the anser be -1/sin(theta)
because it would be cos^2(theta)-1/ sin^2(theta) and then simnplify
"so wwould the anser be -1/sin(theta)
because it would be cos^2(theta)-1/ sin^2(theta) and then simnplify"
It is unfortunate that insufficient use of parentheses leads to mathematical errors.
The expression is correct if we insert parentheses at the right places.
Instead of cos^2(theta)-1/ sin^2(theta), it should be
(cos^2(theta)-1)/ sin^2(theta)
= -sin^2(theta)/sin^2(theta)
= -1
To find the value of cot^2(theta) - csc^2(theta), we need to evaluate the expression for all values of theta for which the expression is defined.
First, let's define the terms cot(theta) and csc(theta):
cot(theta) is the reciprocal of the tangent function, defined as cot(theta) = cos(theta)/sin(theta).
csc(theta) is the reciprocal of the sine function, defined as csc(theta) = 1/sin(theta).
Now, substituting these definitions into the expression, we have:
cot^2(theta) - csc^2(theta) = (cos(theta)/sin(theta))^2 - (1/sin(theta))^2
To simplify this expression, we need to use trigonometric identities. The Pythagorean identity is particularly useful in this case:
sin^2(theta) + cos^2(theta) = 1
Rearranging this equation, we have:
1 = cos^2(theta)/sin^2(theta) + 1/sin^2(theta)
Now, let's substitute this result back into our expression:
cot^2(theta) - csc^2(theta) = (cos(theta)/sin(theta))^2 - (1/sin(theta))^2
= (cos^2(theta)/sin^2(theta)) - 1/(sin^2(theta))
= (cos^2(theta) - 1)/(sin^2(theta))
We know that 1 - sin^2(theta) = cos^2(theta), so we can simplify the expression further:
cot^2(theta) - csc^2(theta) = (1 - sin^2(theta))/(sin^2(theta))
= cos^2(theta)/sin^2(theta)
= cot^2(theta)
Therefore, for all values of theta for which the expression is defined, the value of cot^2(theta) - csc^2(theta) is equal to cot^2(theta).