I need help with some trig problems?

Question

Simplify: (please do step by step)

1) ( ------ is division bar)

sinx 1 cosx
---------- + -------------
1 + cosx sinx


2)

cosxsin^2 - cosx
-------------------------
cosxcotx


( and just so i dont get diferent answers, 1. is cosx and 2. is
-cosxsinx but i cant figure out how to do it

for clarification:
in 1) its sinx divided by 1+cosx and its added to cosx divided by sinx

Answer 1

1) Presuming you mean

sin(x)/[1 + cos(x)] + cos(x)/sin(x)

What we have to do is put this under a common denominator of
[1 + cos(x)][sin(x)]

{ sin(x)sin(x) + cos(x)[1 + cos(x)] } / { [1 + cos(x)][sin(x)] }

Expand the numerator,

{ sin^2(x) + cos(x) + cos^2(x) } / { [1 + cos(x)] [sin(x)] }

We can change sin^2(x) + cos^2(x) to 1.

{ 1 + cos(x) } / { [1 + cos(x)] [sin(x)] }

Note the common term on the top and bottom that cancel out. This leaves us with

1/sin(x)

Which, by definition, is equal to

csc(x)

2) [cos(x)sin^2(x) - cos(x)] / [cos(x)cot(x)]

Factor out a cos(x) on the top.

cos(x) [sin^2(x) - 1] / [cos(x)cot(x)]

We get a cancellation.

[sin^2(x) - 1] / cot(x)

Change cot(x) into cos(x)/sin(x).

[sin^2(x) - 1] / [cos(x)/sin(x)]

Multiply top and bottom by sin(x).

sin(x) [sin^2(x) - 1] / cos(x)

What we're going to do now is use what's known as the "negative 1" technique. This involves factoring out a -1 to change something of the form (a - b) into (-1) (b - a). We're going to do this with sin^2(x) - 1.

{ sin(x) [-1] [1 - sin^2(x)] } / cos(x)

Use the identity 1 - sin^2(x) = cos^2(x)

{ sin(x) [-1] cos^2(x) } / cos(x)

The bottom gets cancelled by the cos^2(x) on the top, reducing cos^2(x) by one power.

sin(x) (-1) cos(x)

Simpilfying, we get

-sin(x)cos(x)

This is actually simple enough. If you wish NOT to see something a bit more advanced, stop reading now!


****

If we really wanted to get fancy, we can use the double angle identity sin(2x) = 2sin(x)cos(x), which implies
(1/2)sin(2x) = sin(x)cos(x), so

-sin(x)cos(x)

is the same as

(-1/2)sin(2x)

Answer 2

Your writing is a little confusing because the font isn't evenly spaced, but looking at the source code of the page I think you mean

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

In the first one, multiply the top and bottom of the first term by (1-cos(x)) and you get:

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

Since (cos(x))^2 + (sin(x))^2 = 1, this becomes

(sin(x)*(1-cos(x)) / (sin(x))^2) + cos(x)/sin(x) =
(sin(x) - sin(x)cos(x)) / (sin(x))^2) + cos(x)/sin(x) =
sin(x)/(sin(x))^2 - sin(x)cos(x)/(sin(x))^2) + cos(x)/sin(x) =
1/sin(x) - cos(x)/sin(x) + cos(x)/sin(x) =
1/sin(x) = csc(x)

3) In the second one, divide both the top and bottom by cos(x). (We'll assume cos(x) doesn't equal zero)

(cos(x)(sin(x))^2 - cos(x)) / cos(x)cot(x)
((sin(x))^2 - 1) / cot(x)

Use the trig identity mentioned earlier and you get

-cos(x))^2 / cot(x)

But cot(x) = cos(x) / sin(x), so this is

-cos(x))^2 * (sin(x) / cos(x)) =
-cos(x)sin(x)

You could also use the rule sin(2x) = 2sin(x)cos(x) to get

(-1/2)sin(2x)