SOLUTION: Prove that cos^2 theta / (cot^2 theta

SOLUTION: Prove that cos^2 theta / (cot^2 theta - cos^2 theta) = 3

Question 239784: Prove that
cos^2 theta / (cot^2 theta - cos^2 theta) = 3
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Before we start a few of notes:

Algebra.com's formula software for some reason does not "do" theta. So I will be using just "t" instead of theta.
Algebra.com's formula software does not handle powers of functions well. You may see some multiplication symbols, "*", between the function and the arguments. These multiplication symbols are not correct and should not be there.
There are two kinds of equations in Trig.:
- Identities which are equations which are true for all possible angles. These are the equations you "prove".
- Conditional equations which are true for only some angles. These equations you solve to find the angles which make the equation true.
The equation you have in this problem is a conditional equation not an identity. So we will not prove it. We will solve it.

Whether you are proving an identity or trying to solve a conditional equation, if the path is not clear to you it is often a good idea to change other Trig functions into sin's and cos's. This is where we will start. Since cot(t) = cos(t)/sin(t)

becomes, after replacing :

Since every term in both the numerator and denominator of the fraction has in it, the fraction will simplify greatly if we multiply the numerator and denominator by its reciprocal:

All the 's cancel leaving:

Now we can further simplify the left side by multiplying the numerator and denominator by :

which gives us:

The denominator should look familiar. It is :

The left side is :

We now have the equation in a form we can solve. (Up to this point, all we've been doing is "playing around" with the equation, using Algebra and Trig. properties/identities, until we find an equation we can solve.)

If then
or
If you know your special angle values well you will recognize that the reference angle is 60 degrees or radians. Since theta is normally used for angles measured in degrees, we will use 60 degrees. So the solution is:
or
Notes about the solution:

The "+ 180n"
- Trig functions are periodic and so when finding angles, there will be an infinite number of solutions. And our solution should reflect all the angles which fit. The "+ 180n" is used as a way to specify this infinite set of angles.
- The "n" can be any integer.
- The "180" is used because the period of the tan function is 180 degrees. (For sin, cos, sec and csc we would use 360.)
The "60". Any angle with a tan of could be used. 60 is the most "obvious".
The "-60". Any angle with a tan of could be used. -60, 120 or 300 are probably the most "obvious".

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