Question

Write the trigonometric expression in terms of sine and cosine, and then simplify. sin(u) cot(u) cos(u)

Answer 1

To rewrite the trigonometric expression in terms of sine and cosine, we can use trigonometric identities to express the expression in a more simplified form.

Step-by-step explanation:

To write the trigonometric expression in terms of sine and cosine and simplify, we can use the following trigonometric identities:

1. sin(u) = sin(u)
2. cot(u) = 1/tan(u)
3. cos(u) = √(1-sin²(u))

Using these identities, we can rewrite the expression as:

sin(u) * (1/tan(u)) * √(1-sin²(u))

Then, we can simplify further if needed.

Answer 2

`sin(u) cot(u) cos(u) = cos^(2)(u)`

Step-by-step explanation:

`sin(u) cot(u) cos(u)`

First, let us simplify cot(u) as follows:

cot (u) =
`(1)/(tan(u))`

also,
`tan (u) = (sin (u))/(cos(u))`

∴
`(1)/(tan(u)) = (1)/((sin(u))/(cos(u)) ) = (cos(u))/(sin(u))`

Hence the original expression becomes:

`sin(u).(cos(u))/(sin(u)) .cos(u)`

Next, sin(u) will cancel each other out, leaving the expression below:

`cos(u) . cos(u) = cos^(2) (u)`

hence:

`sin(u) cot(u) cos(u) = cos^(2)(u)`

I also found a similar expression with a plus (+) sign after the "sin(u)" online, and if this was your question, the solution will be as follows:

sin(u)+ cot(u) cos(u)

`sin(u) + (cos(u))/(sin(u)) . cos (u)`

`= sin(u) + (cos^(2) (u))/(sin(u))`

`sin(u).(sin(u))/(sin(u)) + (cos^(2)(u) )/(sin(u)) \\` (note that
`(sin(u))/(sin(u)) = 1`, hence multiplying it with sin(u) does not change anything in the expression.)

`(sin^(2) (u))/(sin(u)) + (cos^(2)(u) )/(sin(u)) = (sin^(2)(u) + cos^(2)(u) )/(sin(u))`

Now the relationship sin²(u) + cos²(u) = 1

Therefore:

`(sin^(2)(u) + cos^(2)(u) )/(sin(u)) = (1)/(sin(u))`

Hence,
`sin(u)+ cot(u) cos(u) = (1)/(sin(u))`