Mathcad

I think what you are asking about is the trig identity cos^2 (x) + sin^2(x) =1.

(What you have written just cancels down to 1=1.)

To see that the above identity is true,draw yourself a right-angled triangle,name the sides a,b and c with a hypotenuse,b adjacent and c opposite . Now define sin and cos from these and use Pythagoras. 

With regard to validity and domain,the identity is true for all angles.

The number pi was discovered by the ancient Greek mathematicians who were fascinated by the properties of the circle. They discovered that if they measured the area of any circle and divided it by its own radius squared,they alway got the same number.They called it pi. They also discovered that if they measured the circumference of any circle and divided it by its diameter ,they again got this number. Because these  are constant relationships between circles,their areas and diameters (and radii) they are fundaments of mathematics,and pi crops up almost everywhere. It was (and still is)essential in calculations for space exploration for example,and the value of pi has been calculated to hundreds of decimal places by NASA. (How did the Greeks measure the area of a circle?- they drew lots of tiny squares inside the circle and added up all the areas of the squares.The tinier they made the squares,the more accurate the answer they got.) Have a go and see for yourself.

correction to last bit,typo error-

the equals sign should be a plus sign where it reads cos(ln4)=isin(ln4).Should read 

cos(ln4)+isin(ln4)

Sorry!

Answer,then I'll explain.

4^i =cos(ln4)+isin(ln4).

Why? Recall that I can write any number I please,let's say X as always,as

X=e^lnX because e^x and lnX are inverses. So I will write 4^i as e^ln(4^i).

21sin10. The force component along the slope is always mg sin theta where theta is the angle of the slope,m is the mass of the object and g is the force of gravity.

Division by infinity is not defined,in the same way that division by zero is not defined.

Why? A division sign is a question, for example 12/3 means 'what number do I multiply 3 by to get 12?' and the answer is 4. If I write 12/infinity the question is 'what do I multiply infinity by to get 12?' well,it certainly isn't zero because zero times infinity is still zero,not 12.

You can say that the larger a denominator gets,the closer the whole fraction tends to zero,but you need to be careful. If the denominator TENDS to infinity,the fraction tends to zero,but this is not the same as actual division by infinity,which you cannot do.

If you divide a volume in cm^3 by cm the number you get will be in cm^2.

Here's why: Let's take 1000cm^3/100cm for example

Numerator is really (10xcm)x(10Xcm)x(10xcm) =(10x10x10)x(cmxcmxcm)

Denomitor is                                                    100xcm

the dimensions ,that is the cm,cancel down just as the numbers do,so the cm on the denominator cancels out one cm on the numerator,leaving you with the answer 10cm^2.

Similarly,if you divided a volume in cm^3 by cm^2,your answer would be in cm. The thing to realise here is that you mutiply and cancel  dimensions just as you do with  numbers. 

2and3/5 is 13/5.       13/5 x 1/6 =13/30. Just multiply the denominators.

(2= 10/5, add 3/5 to get 13/5.)  because 10/5 +3/5 = (10 +3)/5.  If you have two (or more) fractions with the same denominator -that's the number on the bottom -then you can just add the numerators. Why?

Think this way, one lemon plus two lemons is three lemons. If I  say 'one fifth plus two fifths is three fifths' I'm just adding 'fifths'.

900/2700=0.333333333333333 etc.Now multiply by 100. Gradient is approx 33.33%,or one in three. (Because 900/2700 is just a third.)

round up 479 to 500 giving you 11.25.