Alan

csc = 1/sin and sec = 1/cos so the left-hand side of the equality can be written as:

csc²x + sec²x = 1/sin²x + 1/cos²x

Collect over the common denominator obtained by multiplying sin²x and cos²x together and use the fact that cos² + sin² = 1 .

csc²x + sec²x = 1/sin²x + 1/cos²x = (cos²x + sin²x)/(sin²x*cos²x) = 1/(sin²x*cos²x) = csc²x*sec²x

The final expression is the same as the right-hand side of the original equality.

The graph below shows the cumulative probability (in %) against score for a normal distribution with mean score 100 and standard deviation 15.

The dotted lines mark out scores of 70, 85, 115 and 130.

1. Looks like about 2% of scores are less than 70 (follow the black dotted line from the score of 70 up to the red line and then across to the probability axis), and about 98% in total are below 130. This means that about 98% - 2%, or 96% are between 70 and 130.

Now have a go at the other parts of the question yourself.

pi is bigger than 3 but less than 4, so I would say that it is finite!  However, it does have an infinite decimal expansion.

@reinout  Not all messages are phantom - have a look!

When it says for all a>0, it has already specified that a must be less than 1/4, so I think it means in the range 0<a<1/4.  I might be wrong about this as, for the (r,s) solution it explicitly gives the range as 0 to 3/16.

I calculated the iterates for starting values of (0, 0) and a few different a values (b = 1/2 in all cases).  They look like this:

Where succesive iterates are not converging they are clearly oscillating between two values.

Hope this helps rather than confuses!

(Edited to upload the correct figures!)

Yes, it's functions that have domains and ranges.

The answer is just $$Volume = \frac{4}{3}\pi (\frac{h}{2})^3$$

For part a) just plug in 12 cm for h.

It is possible to do this the hard way, by calculating the bead volume for a non-zero radius hole, and, strictly one should do this in order to prove that the bead volume is independent of the hole radius.  However, since the question here specifies that the volume is independent of hole radius, why make life difficult?!! 

With β = 10*log(l/l_o)  then l = l_o*10^(β/10) 

a) l_conv = l_o*10^(60/10) or l_conv = l_o10⁶       l_pain = l_o*10^(130/10) or l_pain = l_o*10¹³

Therefore l_pain/l_conv = 10¹³/10⁶ = 10^13-6 = 10⁷ .  That is, the intensity at the pain threshold is ten million times greater than at conversation level.

See if you can now do part b)

The binomial theorem (for a cubic expansion) says:

(a + b)³ = a³ + 3a²b +3ab² + b³ 

If you substitute x for a, and 7y for b in this you should get the result shown by kitty<3. 

The following graph should help with part 1).   (Hint: you can't have values of x that make the functions infinite).

For part 2) note that csc = 1/sin and sec = 1/cos 

For part 3) multiply through by sin²x*cos²x (and remember that sin² + cos² = 1)