# Trigonometry and Identities

## Homework Statement

sin^2x + 4sinx +4 / sinx + 2 = sinx +2

## The Attempt at a Solution

L.S = sin^2x + 4sinx +4 / sinx + 2
=1-cos^2+4(sinx + 1) / sinx +2

Not sure where to go from there.
Not sure if I was even supposed to factor out the 4?

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SammyS
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## Homework Statement

sin^2x + 4sinx +4 / sinx + 2 = sinx +2

## The Attempt at a Solution

L.S = sin^2x + 4sinx +4 / sinx + 2
=1-cos^2+4(sinx + 1) / sinx +2

Not sure where to go from there.
Not sure if I was even supposed to factor out the 4?
Please enclose the entirety of any numerator and/or denominator in parentheses.

Please enclose the entirety of any numerator and/or denominator in parentheses.
(Sin^2x + 4sinx + 4) / (sinx + 2) = sinx + 2

SammyS
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(Sin^2x + 4sinx + 4) / (sinx + 2) = sinx + 2
Factor the numerator.

Factor the numerator.
Thank you, didn't catch that.

SammyS
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Thank you, didn't catch that.
So, what do you get ?

So, what do you get ?
((Sinx + 2)(Sinx + 2)) / (Sinx + 2)

Then you cancel one from top and bottom to get: Sinx + 2.

((Sinx + 2)(Sinx + 2)) / (Sinx + 2)

Then you cancel one from top and bottom to get: Sinx + 2.
Yes, but it is a tiny bit more complicated. Here's something to think about:

1) Why doesn't the following equality hold for all $x$:

$$\frac{(x+2)(x+2)}{x+2} = x+2$$

2) Why is this no problem with the question in the OP?

Yes, but it is a tiny bit more complicated. Here's something to think about:

1) Why doesn't the following equality hold for all $x$:

$$\frac{(x+2)(x+2)}{x+2} = x+2$$

2) Why is this no problem with the question in the OP?
((Sinx + 2)(Sinx + 2)) you then take reciprocal of denominator and multiply it by the numerator, and that it is when you cancel them out?

Math_QED
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2019 Award
Can you always divide out common factors from numerator and denumerator? For example, can you always say that (cosx-1)(cosx + 1)/(cosx - 1) = cosx + 1?

Why can/can't you say that? And what about your expression, those are things you have to think about!

Yes, but it is a tiny bit more complicated. Here's something to think about:

1) Why doesn't the following equality hold for all $x$:

$$\frac{(x+2)(x+2)}{x+2} = x+2$$

2) Why is this no problem with the question in the OP?
To give you a hint, what happens if we plug in -2 for x? Pay attention to the denominator.

Mark44
Mentor
Yes, but it is a tiny bit more complicated. Here's something to think about:

1) Why doesn't the following equality hold for all $x$:

$$\frac{(x+2)(x+2)}{x+2} = x+2$$

2) Why is this no problem with the question in the OP?
Veronica Oles said:
((Sinx + 2)(Sinx + 2)) you then take reciprocal of denominator and multiply it by the numerator, and that it is when you cancel them out?
micromass asked two questions. You didn't respond to his first question, and your answer to the second question doesn't address why $\frac{(\sin x+2)(\sin x+2)}{\sin x+2} = \sin x+2$ is always true, regardless of the value of x.