document.write( "Question 239784: Prove that
\n" ); document.write( "cos^2 theta / (cot^2 theta - cos^2 theta) = 3 \n" ); document.write( "
Algebra.Com's Answer #175891 by jsmallt9(3758)\"\" \"About 
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.

\n" ); document.write( "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)
\n" ); document.write( "\"cos%5E2%28t%29%2F+%28cot%5E2%28t%29+-+cos%5E2%28t%29%29+=+3\"
\n" ); document.write( "becomes, after replacing \"cot%5E2%28t%29\":
\n" ); document.write( "\"cos%5E2+%28t%29%2F+%28%28cos%5E2+%28t%29%2Fsin%5E2%28t%29%29+-+cos%5E2+%28t%29%29+=+3\"
\n" ); document.write( "Since every term in both the numerator and denominator of the fraction has \"cos%5E2%28t%29\" in it, the fraction will simplify greatly if we multiply the numerator and denominator by its reciprocal:
\n" ); document.write( "
\n" ); document.write( "All the \"cos%5E2%28t%29\"'s cancel leaving:
\n" ); document.write( "\"1%2F%281%2Fsin%5E2%28t%29+-+1%29+=+3\"
\n" ); document.write( "Now we can further simplify the left side by multiplying the numerator and denominator by \"sin%5E2%28t%29\":
\n" ); document.write( "\"%281%2F%281%2Fsin%5E2%28t%29%29+-+1%29%28sin%5E2%28t%29%2Fsin%5E2%28t%29%29+=+3\"
\n" ); document.write( "which gives us:
\n" ); document.write( "\"sin%5E2%28t%29%2F%281-sin%5E2%28t%29%29+=+3\"
\n" ); document.write( "The denominator should look familiar. It is \"cos%5E2%28t%29\":
\n" ); document.write( "\"sin%5E2%28t%29%2Fcos%5E2%28t%29+=+3\"
\n" ); document.write( "The left side is \"tan%5E2%28t%29\":
\n" ); document.write( "\"tan%5E2%28t%29+=+3\"
\n" ); document.write( "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.)

\n" ); document.write( "If \"tan%5E2%28t%29+=+3\" then
\n" ); document.write( "\"tan%28t%29+=+sqrt%283%29\" or \"tan%28t%29+=+-sqrt%283%29\"
\n" ); document.write( "If you know your special angle values well you will recognize that the reference angle is 60 degrees or \"pi%2F3\" radians. Since theta is normally used for angles measured in degrees, we will use 60 degrees. So the solution is:
\n" ); document.write( "\"t+=+60+%2B+180n\" or \"t+=+-60+%2B+180n\"
\n" ); document.write( "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 \"sqrt%283%29\" could be used. 60 is the most \"obvious\".
  • The \"-60\". Any angle with a tan of \"-sqrt%283%29\" could be used. -60, 120 or 300 are probably the most \"obvious\".
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