Can (sec x - cosec x) / (tan x - cot x) be simplified further?

Given the expression ( sec x - csec x ) / (tan x - cot
x)

We need to simplify.

We
will use trigonometric identities to simplify.

We know
that:

sec x = 1/cos x

csec x =
1/sin x

tan x = sinx/cosx

cot
x = cos x/ sin x

We will substitute into the
expression.

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

==>[ ( sin x - cos x) / sinx*cosx]
/ [ (sin^2 x - cos^2 x)/ cosx*sinx

We will reduce
similar:

==> (sinx - cos x) / (sin^2 x - cos^2
x)

Now we will simplify the
denominator.

==> (sin x - cos x) / (sin x - cos
x)(sinx + cos x)

Reduce similar
terms.

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

Then the expression ( sec x - csec
x)/(tanx - cot x) can be written as : 1/(sin x + cos
x)

the coifficent of {r-1}th ,r th ,{r-1}th term in expansion of {x+1}n are in the ratio 1:3:5,find n&r.

The problem provides the equations that relates the
binomial coefficients of the `(r-1),r ` and `(r+1)` terms, such
that:

`(C_n^r)/(C_n^(r+1)) = 1/3 => (C_n^(r+1)) =
3(C_n^r)`

Using the factorial formulas
yields:

`(n!)/((r+1)!(n - r - 1)!) = (3n!)/((r!)(n - r)!)`

Since `(r+1)! = r!(r+1)` and `(n-r)! = (n-r-1)!(n-r)`
yields:

`1/(r+1) = 3/(n-r) => 3(r+1) = n-r`

`(C_n^(r+1))/(C_n^(r+2)) =
3/5`

`3(C_n^(r+2)) = 5(C_n^(r+1))`

`3(n!)/((r+2)!(n - r - 2!) = 5(n!)/((r+1)!(n - r -
1)!)`

`3/(r+2) = 5/(n-r-1 )=> 5(r+2) = 3(n-r-1)`

You need to solve for n and r the system of simultaneous
equations, such that:

`{(3(r+1) = n-r),(5(r+2) =
3(n-r-1)):}`

`=> {(4r - n = -3),(8r - 3n = -13):}
=> {(-8r + 2n = 6),(8r - 3n = -13):} => -n = -7 => n = 7 r =
(n-3)/4 => r = (7-3)/4 => r =
1`

Hence, evaluating r and n, under the given
conditions, yields `r = 1` and `n = 7` .

What main events happen in Chapters 34-36 in The Adventures of Tom Sawyer by Mark Twain?

These are the final short chapters of the Twain classic.
After Tom and Huck return from finding the treasure, they also find that the Widow
Douglas has a house full of people and some news of her own: She is going to adopt Huck.
But before she can get Huck's response, Tom blurts out that they are rich. Tom brings in
the sacks of gold and when they count the money, they find that it totals $12,000. Tom
and Huck became the biggest celebrities in the town. The Widow Douglas and Aunt Polly
put their boys' money in the bank "at six per cent," and even gave them a gigantic
allowance (for the time!) of $1 per day. But such life was not for Huck, and the two
decided to get their gang back together--with an initiation at
midnight.

In Act III, scene 2, why may the establishment of Claudius's guilt be considered the crisis of the revenge plot?

The crisis of a drama usually proceeds and leads to the
climax.  In Shakespeare's Hamlet, the proof that Claudius is guilty
leads to Hamlet's decision to not kill Claudius while he's at prayer--and that is the
climax of the play.

Hamlet, until he sees Claudius's
reaction to the "play within the play," isn't entirely sure Claudius is guilty.  He has
no real proof--only the word of a ghost, who, he says in Act 2.2.565-572, could be a
devil trying to deceive him (as, by the way, the witches do to Macbeth in his play of
the same name).  Hamlet needs proof.  He is too reasonable to act like Fortinbras or
Laertes and just jump into revenge without thinking it
through.

The king's reaction to the murder scene in the
play gives Hamlet the proof he needs, though, and he sets off to kill the king.  He gets
an opportunity but decides not to take it.  Why?  Because he thinks Claudius is
confessing (he isn't, but Hamlet doesn't know that), and killing him immediately after
he confesses his sins would send Claudius straight to heaven.  And Hamlet doesn't want
to send Claudius to heaven, not when his father is suffering in a purgatory-like state,
and when Hamlet might be sent to hell because he kills
Claudius. 

The problem is, though, that when Hamlet decides
not to kill Claudius because he doesn't want to contribute to his salvation, he is
playing God.  Salvation is God's business, not Hamlet's.  Hamlet is messing where he
shouldn't be messing.

The result--you see it in Act 5:  the
sight Fortinbras says doesn't belong in a castle, only on a battlefield.  Death
everywhere. 

When Hamlet walks away from his rightful
revenge, by playing God, he dooms himself and so many others.  This is the climax. 
His receiving proof of Claudius's death could be considered the crisis, and Hamlet's
refusal to kill Claudius while the king's at prayer is the climax.  One leads to the
other.  

Explain Shakespeare's pun on the word "arms" in Act 5 of Hamlet?

Even though the setting of this scene is a graveyard,
Shakespeare interjects some humor in the scene in order to relieve the tension of the
previous act where Claudius and Laertes plot Hamlet's death and the coming end of the
play where there will be inevitable deaths of at least some if not all of the main
characters.  There is a kind of "dark humor" in the fact that funniest person in the
play is the man who digs graves for a living.

The pun you
are referencing is in a conversation between the two gravediggers.  A pun is a play
onwords when two different meanings of a word are used together.  The first speaker says
that gravediggers hold up (continue) Adam's profession.  He is referring to Adam (and
Eve) the first humans.  The second clown asks, "Was he a gentleman?" The pun starts when
the first clown replies "The first that ever bore arms."  He means had arms with which
to work, but the second clown retorts, "Why he had none."  He is using the second
definition of arms -- meaning weapons.  Clearly, Adam didn't have any weapons.  The puns
is extended when the first clown comes back to "his" definition and asks, "Are you a
heathen? The scripture says Adam digged.  Could he dig without arms?"  This short scene
is a clever and silly respite from an otherwise heavy and depressing
situation.

Explain what happened to Japanese-Americans once they were freed from the camps.

In general, the Japanese Americans who were released from
the camps were left to fend for themselves.  They had to figure out what to do and how
to put their lives back together after they were
released.

Some of the Japanese Americans went back to their
homes and tried to go back to their former lives.  In Farewell to
Manzanar, for example, Jeanne Wakatsuki Houston tells about her family's
attempt to do this.  They were unsuccessul and eventually had to move to a different
town as her father tried to find a different job.

Other
Japanese Americans settled in places where they had not lived before.  My area of
Central Washington has a number of Japanese American families who originally came here
out of the internment camps.

Some few Japanese went back to
Japan.

Overall, then, there was nothing official done with
the Japanese American internees.  They were simply let out of the camps and left to fend
for themselves.

What are some symbols in "A Very Old Man with Enormous Wings?"

One symbol is the infestation of crabs in Pelayo's house.
During the strong rains, the land crabs hide indoors so they won't drown; Pelayo kills
them and throws them into the water. The crabs symbolize both Pelayo's poverty -- he
can't afford a better house -- and the instinct of all animals to survive under
pressure. When the angel shows up, it is lying in the mud, trying to move but stuck.
Just like the crabs, the angel has the instinct to survive, but it is too
old:

...before
going to bed [Pelayo] dragged him out of the mud and locked him up with the hens in the
wire chicken coop.
(Márquez, "A Very Old Man with Enormous Wings,"
salvoblue.homestead.com)

Although
they do not treat it well, Pelayo and his wife offer the angel sanctuary to live through
the winter; its instincts, if it sought them out specifically, were sound. After gaining
money, Pelayo builds a new house, this time with barriers to keep the crabs out; the
symbol of the crabs has been replaced by the angel, which now lives in the unchanged
chicken coop, representing the poverty that Pelayo and his wife used to live in. The
crabs, therefore, have served their purpose, and are no longer
necessary.