Cos^2(theta)-3csc(theta)+2 divided by csc(theta)-1
I hope you mean
(csc^Ø - 3cscØ + 2)/(cscØ - 1)
if so, then
(cscØ - 1)(cscØ - 2)/(cscØ-1)
= cscØ - 2
It's Cos^2(theta) is that the same as csc^(theta)?
no
csc^(theta) is meaningless notation. Theta is never an exponent (although it could be). You would still need to specify an angle for which you are taking the cosecant.
cos^2(theta) means the same thing as
[cos(theta)]^2
To simplify the expression (cos^2(theta) - 3csc(theta) + 2) / (csc(theta) - 1), we can follow these steps:
Step 1: Identify any trigonometric identities that can be used to simplify the expression.
Step 2: Convert all trigonometric functions to sine and cosine using the identities:
- csc(theta) = 1/sin(theta)
- cos^2(theta) = 1 - sin^2(theta)
Step 3: Apply algebraic simplifications to the numerator and denominator.
Let's go through each step in detail:
Step 1: Identify any trigonometric identities:
We can see that the expression involves cos^2(theta), csc(theta), sin(theta), and cos(theta). We can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 to simplify the expression further.
Step 2: Convert all trigonometric functions to sine and cosine:
Replace csc(theta) with 1/sin(theta) and cos^2(theta) with 1 - sin^2(theta), which gives us:
((1 - sin^2(theta)) - 3(1/sin(theta)) + 2) / ((1/sin(theta)) - 1)
Step 3: Apply algebraic simplifications:
To simplify the numerator, distribute the -3 to get:
(1 - sin^2(theta)) - (3/sin(theta)) + 2
Combine like terms:
- sin^2(theta) - (3/sin(theta)) + 3
To simplify the denominator, use the common denominator of sin(theta) to write it as:
(1/sin(theta)) - (sin(theta)/sin(theta))
Combine the fractions:
(1 - sin^2(theta) - sin(theta))/sin(theta)
Now, let's simplify the numerator:
- sin^2(theta) - (3/sin(theta)) + 3
= -(sin^2(theta) + (3/sin(theta))) + 3
To simplify further, combine the two terms under a common denominator:
= (-sin^2(theta)*sin(theta) - 3)/sin(theta) + 3
Combine the terms in the numerator:
= (-sin^3(theta) - 3)/sin(theta) + 3
Finally, the simplified expression is:
((-sin^3(theta) - 3)/sin(theta) + 3) / ((1 - sin^2(theta) - sin(theta))/sin(theta))
Please note that this is the simplified form of the expression, but it might still be possible to simplify it further depending on the given value for theta.