Prove the identity: csc x - cot x = sinx/1+cosx

The identity csc x - cot x = sin x/(1+cos x) has to be
proved.

Start from the left hand
side

csc x - cot x

csc x = 1/
sin x and cot x = cos x/ sin x

=> 1/sin x - cos
x/sin x

=> (1 - cos x)/sin
x

=> (1 - cos x)(1 + cos x)/(sin x)*(1 + cos
)

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

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

=> sin x / (1 + cos
)

which is the right  hand
side.

This proves csc x - cot x = sin
x/(1+cos x)

What are the chorus characters in Macbeth?

If you are referring to "chorus characters" as characters
who act as a unified whole (similar to the role of the chorus in Greek plays) then the
three witches would be the chorus characters in Macbeth.  The three
act in unison when giving Macbeth and Banquo the predictions for their future.  They
also continue to plot future events.  When Hecate enters the play, she is angry with all
three witches for conspiring together and telling Macbeth information that he should not
know.  Throughout the play, the witches serve as a force for challenging Macbeth's sense
of morality and righteousness.  So, they serve a similar role in
Macbeth as the chorus does in Greek plays.

Why is the first scene of Macbeth important?

The first scene of Macbeth is
important because it sets the tone of events to come.  In the scene, the three witches
meet and discuss when they will meet again.  Here, the reader learns that a battle will
ensue and that they will come together afterwards.  But they say that the battle will be
both lost and won which is a paradox meaning literally that the battle will be lost by
one side and won by the other.  Figuratively, the statement refers to Macbeth's
future--he will win the "battle" of becoming the King and reaching his ambitious goals;
however, he will lose his family, the respect of his followers, and eventually his life
in the process.  Further, the witches end the scene with another paradox:  "Fair is foul
and foul is fair."  This sets the tone for the play--all will not be as it
seems.

How can it be proved that the sum of angles of a triangle equal 180 degrees.

Let ABC be a triangle.

Exend
AC to X from C.

Draw a line CD frm C which is || to
AB.

Now AB || CD and BC is a
transvalal.

Therefore angle ABC = angle BCD being alternate
angle.

Therefore angle CAB = angle DCX being corresponding
angles.

Therefore angle CAB + angle ABC +angle ACB = Angle
ACB+Angle ABD+Angle DCX. The right side is the sum of angle on the straight line, which
is 180 degree.

This proves that the sum of
the angles of a triangle is 180 degrees.

Why did Marlow change his desire for exploring the Arctic to going to Africa in Heart of Darkness?

If we look at the first section in this incredibly
powerful novella, Marlow recounts to us how as a child, he had a fascination for maps.
He talks about how he would spend hours looking at them and thinking of places he would
like to go. One of them was the North Pole, but he says he will not go there now,
because "the glamour's off." Obviously, now that he is an adult, the childlike
enthusiasm and innocence that made him desire to go there has
abated.

However, what attracted him to Africa was the way
that it was "the biggest, the most blank" out of the continents. However, as Marlow
grows up, so Africa becomes more explored and charted, and rivers and other geographical
forms are recorded on maps. It was no longer a place of "delightful mystery" and had
instead become a place of "darkness." However, there is one river in particular that
seems to draw Marlow and tempt him:

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But there was in it one river especially, a
mighty big river, that you could see on the map, resembling an immense snake uncoiled,
with its head in the sea, its body at rest curving afar over a vast country, and the
tail lost in the depths of the land. And as I looked at the map of it in a shop-window,
it fascinated me as a snake would a bird--a silly little
bird.

Note how the river is
compared to a snake--a symbol of temptation and evil, and it is said to "charm" Marlow
as if he were a bird. It is this that makes Marlow decide to go to Africa and experience
the "darkness" of this place for himself.

What were the conditions in Europe that led to the conquest of the Americas?history

The major condition in Europe that led to the original
conquest of the Americas was peace.  This applies mainly to Spain which, of course, was
the country that was first involved in conquering the New
World.

Spain came to be able to explore because of the
coming of peace.  Up until 1492, the Spanish had been involved in wars against the
Muslims, trying to push them out of the Iberian Peninsula.  But then the Spanish won
that struggle and were at peace.  At that point, they were able to put their money
towards exploration.  Once Columbus "discovered" the New World, the Spanish were able to
send soldiers to conquer who were no longer needed for wars in Spain
itself.

While The Laramie Project is billed as a Verbatim performance, how does part of it differ from the Verbatim form?

I think that there are aspects of Kaufman's work that
represents verbatim theatre.  The most notable of this would be the collection of
interviews from people in Laramie and the interviews of the members of the Tectonic
group themselves.  This represents verbatim theatre because of its docudrama style,
allowing the voices of the people in the setting speak for themselves.  This also
represents verbatim theatre because it allows for a type of historiography present in
the way in which interviews are collected, marking primary source footage in a secondary
source element.  Where I think that Kaufman's work might deviate a bit from the verbatim
format lies in how the work presents life in Laramie and Matt Shepard's life, in
general:

In
order to accurately present the information that he and his troupe had gathered, Kaufman
created the illusion of reality by formatting his play not as a fictional story, but
rather as a reenactment of the interviews. The fictional, or artistic, part of the play
was in how Kaufman pulled all this information together and made it tell a
story.

It is here where
Kaufman differs from true verbatim style, which is more concerned with a historical
recreation.  Kaufman wants the political message of his work to have the greatest
effect.  This means that the construction of Matt Shepard's experiences and his life in
Laramie is done to facilitate a larger discussion of homophobia and fear of "the
other."  Verbatim theatre is more concerned with this recreation, whereas Kaufman's work
seeks to start a much needed discourse.  This is why specific parts of the work can move
away from verbatim style, as Kaufman seeks another form while embracing aspects of
it.