What is the simplified for of the expression cos theta / csc theta - sin theta?
How do you show your work on this?
since cscθ = 1/sinθ,
cosθ/cscθ - sinθ
= cosθ sinθ - sinθ
= sinθ(cosθ-1)
or maybe you meant
cosθ/(cscθ-sinθ)
= cosθ/(1/sinθ - sinθ)
= cosθ sinθ / (1 - sin^2θ)
= cosθ sinθ / cos^2θ
= sinθ/cosθ
= tanθ
parentheses can make a difference!
Well, let's take a closer look at the expression. We have cos theta / csc theta - sin theta. We can simplify this by manipulating the trigonometric identities.
Now, csc theta is the reciprocal of sin theta, so we can rewrite it as 1 / sin theta. Let's substitute that in: cos theta / (1 / sin theta) - sin theta.
Next, let's simplify the expression by multiplying both the numerator and denominator of the fraction by sin theta to get rid of the denominator: (cos theta * sin theta) / 1 - sin^2 theta.
Remember the Pythagorean identity sin^2 theta + cos^2 theta = 1. We can now rewrite the expression as (cos theta * sin theta) / (1 - sin^2 theta) = (cos theta * sin theta) / cos^2 theta.
Finally, we can cancel out the common factor of cos theta in the numerator and denominator, leaving us with: sin theta / cos theta. And this, my friend, is simply the tangent of theta, written as tan theta.
As for showing the work, you can write your steps down similar to what I've done here, using the trigonometric identities and simplifying along the way. Just remember to keep a sense of humor throughout the process!
To simplify the expression cos(theta) / csc(theta) - sin(theta), we can rewrite the trigonometric functions in terms of sine and cosine:
csc(theta) = 1 / sin(theta)
So we can substitute 1 / sin(theta) for csc(theta) in the expression:
cos(theta) / csc(theta) - sin(theta)
= cos(theta) / (1 / sin(theta)) - sin(theta)
= cos(theta) * sin(theta) - sin(theta)
Next, we can factor out sin(theta) as a common term:
= sin(theta) * (cos(theta) - 1)
Therefore, the simplified form of the expression cos(theta) / csc(theta) - sin(theta) is sin(theta) * (cos(theta) - 1).
To show your work, it is important to clearly indicate each step of simplification, as I did above. You can write out each step on separate lines, numbering them to make the steps clear. Additionally, you can use proper notation and symbols to represent the trigonometric functions.
To simplify the expression cos(theta)/csc(theta) - sin(theta), we need to understand the basic trigonometric identities and manipulate the given terms.
1. Start with the expression cos(theta)/csc(theta) - sin(theta).
2. First, simplify csc(theta) by taking the reciprocal of sin(theta). So, csc(theta) = 1/sin(theta).
3. Now substitute the simplified value of csc(theta) into the expression: cos(theta)/(1/sin(theta)) - sin(theta).
4. To divide by a fraction, multiply by its reciprocal. Multiply both the numerator and denominator by sin(theta), which results in:
cos(theta) * sin(theta) - sin(theta)
5. Factor out sin(theta) as a common term: sin(theta) * (cos(theta) - 1).
So, the simplified form of the expression cos(theta)/csc(theta) - sin(theta) is sin(theta) * (cos(theta) - 1).
To show your work, it is important to write down the step-by-step process of simplification, as outlined above. Make sure to highlight the trigonometric identities used and the algebraic manipulation performed.