Mathematical Re...

While simplifying $$\sqrt { \frac { 1-cosx }{ 1+cosx }  } $$, two students got the following two answers A & B.
A)  cosec x - cot x  (B)   $$\frac { 1 }{ cosecx+cotx } $$ What can you say about answers ?

Which of the following proportional is a tautology? 

The logically equivalent proportion of $$p \leftrightarrow q$$ is

If  p, q, r are statements with truth values false, true and false respectively.  then truth value of $$(\sim pv \sim q) v r$$  is 

The converse of the contra positive of $$
\sim p \rightarrow q
 $$ is ........(1)

The contrapositive of $$( p \wedge q ) \Rightarrow r$$ is

Consider the statement :$$"P\left( n \right) :{ n }^{ 2 }-n+41$$ is prime." Then which one of the following is true?

Two pairs of statement are:
p: If a quadrilateral is a rectangle, then its opposite sides are equal.
q: If opposite sides of a quadrilateral are equal, then the quadrilateral is a rectangle.
The combined statement of these pairs using If and only if is:

Let A and B denote the statements
A:$$\cos { \alpha  } +\cos { \beta  } +\cos { \gamma  } =0$$ & B:$$\sin { \alpha  } +\sin { \beta  } +\sin { \gamma  } =0$$
If $$\cos { (\beta -\gamma ) } +\cos { (\gamma -\alpha ) } +\cos { (\alpha -\beta ) } =-\frac { 3 }{ 2 } $$ then

If $$x = 5$$ and $$y = - 2$$ then $$x - 2 y = 9 .$$ The contrapositive of this statement is