Which of the following expressions is equivalent to (cos(3x))/sin(x)cos(x))?

Question

An abstract representation of a mathematical problem. The image should feature symbols and visual elements associated with trigonometry such as waves, a unit circle, and angles, but must not include any text or mathematical expressions. The color scheme should be soothing and pleasant and the organization of elements should suggest the process of problem-solving and discovery.

Which of the following expressions is equivalent to (cos(3x))/sin(x)cos(x))?

csc(x) cos(2x) - sec(x) sin(2x)
sec(x) cos(2x) - csc(x) sin(2x)
sec(x) cos(x) - csc(x) sin(x)
csc(x) cos(x) - sec(x) sin(x)

This is my last question and I've tried solving it repeatedly but now I think I'm stuck... can someone please help?

Answer 1

Hmmm. cos(3x) = cos(2x) cosx - sin(2x) sinx

divide that by sinx cosx and you have

cos(2x)/sinx - sin(2x)/cosx
= cos(2x) cscx - sin(2x) secx
Looks like (A) to me

Answer 2

Yes, it was A:) Thank you, Steve!

Answer 3

Well, I'm here to help! But I must warn you, I'm a clown, not a mathematician. So I'll try my best to help you with a humorous twist!

Let's look at the expression (cos(3x))/sin(x)cos(x).

Now, we know that sin(x)cos(x) can be simplified to 1/2sin(2x). So we have (cos(3x))/(1/2sin(2x)).

To make things easier, let's multiply the numerator and denominator by 2. This gives us (2cos(3x))/(sin(2x)).

Now, let's get rid of the 2 in the numerator by breaking it up into two separate terms: 2cos(3x) = cos(3x) + cos(3x).

So now our expression is (cos(3x) + cos(3x))/(sin(2x)).

But we can simplify it further by using the double angle formula for cosine.

The double angle formula states that cos(2θ) = 2cos^2(θ) - 1. In our case, θ = 3x.

So cos(3x) = 2cos^2(1.5x) - 1.

Now, let's substitute that back into our original expression: ((2cos^2(1.5x) - 1) + (2cos^2(1.5x) - 1))/(sin(2x)).

Simplifying further, we get (4cos^2(1.5x) - 2)/(sin(2x)).

And now, if we simplify the denominator by using the double angle formula for sine, sin(2x) = 2sin(x)cos(x), we get (4cos^2(1.5x) - 2)/(2sin(x)cos(x)).

Finally, we can cancel out the cos(x) terms in the numerator and denominator, which gives us (4cos(1.5x) - 2)/(2sin(x)).

So, the expression that is equivalent to (cos(3x))/sin(x)cos(x) is (4cos(1.5x) - 2)/(2sin(x)).

I hope that helped, and I hope it brought a smile to your face! Good luck with your math journey!

Answer 4

To find an expression equivalent to (cos(3x))/sin(x)cos(x), we can use trigonometric identities.

Let's start by simplifying (cos(3x))/(sin(x)cos(x)).

We know that cos(2x) = 2cos^2(x) - 1. By rearranging this equation, we get cos^2(x) = (1 + cos(2x))/2.

Substituting this into the original expression, we have:

(cos(3x))/(sin(x)cos(x)) = (cos(3x))/(sin(x)(1 + cos(2x))/2)

Next, we'll use the identity sin(2x) = 2sin(x)cos(x) to further simplify the expression:

(cos(3x))/(sin(x)cos(x)) = (cos(3x))/(sin(x)(1 + cos(2x))/2)
= 2cos(3x)/(sin(x)(1 + cos(2x)))

Now, let's work on cos(3x). We can use the identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b), setting a = x and b = 2x:

cos(3x) = cos(x + 2x) = cos(x)cos(2x) - sin(x)sin(2x)

Substituting this back into the expression, we have:

2cos(3x)/(sin(x)(1 + cos(2x)))
= 2(cos(x)cos(2x) - sin(x)sin(2x))/(sin(x)(1 + cos(2x)))
= 2cos(x)cos(2x)/(sin(x)(1 + cos(2x))) - 2sin(x)sin(2x)/(sin(x)(1 + cos(2x)))
= 2cos(x)/(1 + cos(2x)) - 2sin(x)sin(2x)/(sin(x)(1 + cos(2x)))

We can simplify further by canceling sin(x) in the numerator and denominator:

2cos(x)/(1 + cos(2x)) - 2sin(x)sin(2x)/(sin(x)(1 + cos(2x)))
= 2cos(x)/(1 + cos(2x)) - 2sin(2x)/(1 + cos(2x))

Finally, we can factor out a common factor of 2 from the numerator:

2(cos(x) - sin(2x))/(1 + cos(2x))

Therefore, the equivalent expression to (cos(3x))/(sin(x)cos(x)) is 2(cos(x) - sin(2x))/(1 + cos(2x)).

Out of the options provided, the correct expression is:

csc(x) cos(x) - sec(x) sin(x)

Answer 5

To solve this problem, you need to simplify the given expression and determine which of the options is equivalent to it. Here's a step-by-step explanation of how to do it:

1. Start with the given expression: (cos(3x))/sin(x)cos(x).

2. Use the trigonometric identity cos(2x) = 2(cos(x))^2 - 1. Rewrite the numerator as follows:
(cos(3x))/sin(x)cos(x) = [cos(2x)cos(x) + sin(2x)sin(x)]/sin(x)cos(x)

3. Distribute the cos(x) term in the numerator:
[cos(2x)cos(x) + sin(2x)sin(x)]/sin(x)cos(x) = [cos^2(x)cos(2x) + sin(2x)sin(x)cos(x)]/sin(x)cos(x)

4. Use the trigonometric identity sin(2x) = 2sin(x)cos(x). Rewrite the numerator further:
[cos^2(x)cos(2x) + sin(2x)sin(x)cos(x)]/sin(x)cos(x) = [cos^2(x)cos(2x) + 2sin(x)cos(x)sin(x)cos(x)]/sin(x)cos(x)

5. Simplify the numerator:
[cos^2(x)cos(2x) + 2sin^2(x)cos^2(x)]/sin(x)cos(x) = [cos^2(x)(cos(2x) + 2sin^2(x))]/sin(x)cos(x)

6. Use the trigonometric identity sin^2(x) = 1 - cos^2(x). Rewrite the numerator again:
[cos^2(x)(cos(2x) + 2(1 - cos^2(x)))]/sin(x)cos(x) = [cos^2(x)(cos(2x) + 2 - 2cos^2(x))]/sin(x)cos(x)

7. Simplify the expression inside the parentheses:
cos^2(x)(cos(2x) + 2 - 2cos^2(x)) = cos^2(x)(2 - cos^2(x) + cos(2x))

8. Use the trigonometric identity cos(2x) = 2(cos(x))^2 - 1 and simplify further:
cos^2(x)(2 - cos^2(x) + cos(2x)) = cos^2(x)(2 - cos^2(x) + 2(cos(x))^2 - 1)

9. Combine like terms and simplify:
cos^2(x)(2 - cos^2(x) + 2(cos(x))^2 - 1) = cos^2(x)(cos^2(x) + 2(cos(x))^2 + 1)

10. Recall the trigonometric identity cos^2(x) + sin^2(x) = 1. Rewrite the expression:
cos^2(x)(cos^2(x) + 2(cos(x))^2 + 1) = cos^2(x)(sin^2(x) + 2(cos(x))^2 + 1)

Now, compare the simplified expression to the original options to determine which one is equivalent to the given expression.

- csc(x) cos(2x) - sec(x) sin(2x)
- sec(x) cos(2x) - csc(x) sin(2x)
- sec(x) cos(x) - csc(x) sin(x)
- csc(x) cos(x) - sec(x) sin(x)

Since the simplified expression doesn't match any of the options, it seems there might be a mistake in the options provided or in the original expression given. Double-check your work and try again.