document.write( "Question 595159: Use pythagorgean identities to write the expression as an integer.\r
\n" ); document.write( "\n" ); document.write( "tan^2 6beta - sec^2 6beta\r
\n" ); document.write( "\n" ); document.write( "6 tan^2 beta - 6 sec^2 beta\r
\n" ); document.write( "\n" ); document.write( "2sin^2 (theta/4) + 2cos^2 (theta/4) \n" ); document.write( "
Algebra.Com's Answer #377132 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
It takes time but eventually you should learn that the variables in all the properties/identities you learn are simply place-holders. When we learn that
\n" ); document.write( "\"sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1\"
\n" ); document.write( "we are learning that sin squared of anything + cos squared of that same thing add up to a 1:
\n" ); document.write( "\"sin%5E2%28y%29+%2B+cos%5E2%28y%29+=+1\"
\n" ); document.write( "\"sin%5E2%28x%2B9%29+%2B+cos%5E2%28x%2B9%29+=+1\"
\n" ); document.write( "\"sin%5E2%28xy%29+%2B+cos%5E2%28xy%29+=+1\"
\n" ); document.write( "\"sin%5E2%2823x%29+%2B+cos%5E2%2823x%29+=+1\"
\n" ); document.write( "etc.

\n" ); document.write( "The same goes for all the others:
\n" ); document.write( "sin(2x) = 2sin(x)cos(x)
\n" ); document.write( "sin(14x) = 2sin(7x)cos(7x)
\n" ); document.write( "sin(200q) = 2sin(100q)cos(100q)
\n" ); document.write( "etc.
\n" ); document.write( "(All that matters here is that the argument on the left is twice as much as the arguments on the right.)

\n" ); document.write( "Once we get this \"place-holder\" idea, problems like yours become very easy.
\n" ); document.write( "\"tan%5E2%286beta%29+-+sec%5E2%286beta%29\"
\n" ); document.write( "The Pythagorean identity that involves tan and sec is:
\n" ); document.write( "\"1+%2B+tan%5E2%28x%29+=+sec%5E2%28x%29\"
\n" ); document.write( "With a little algebra we can work this around to look like your expression. Subtracting 1 and \"sec%5E2%28x%29\" from each side we get:
\n" ); document.write( "\"tan%5E2%28x%29+-+sec%5E2%28x%29+=+-1\"
\n" ); document.write( "This tells us that, as long as the two arguments are equal, tan squared minus sec squared is always equal to -1! So your expression is equal to -1, too!

\n" ); document.write( "\"6tan%5E2%28beta%29+-+6sec%5E2%28beta%29\"
\n" ); document.write( "Here we start by factoring out a 6:
\n" ); document.write( "\"6%28tan%5E2%28beta%29+-+sec%5E2%28beta%29%29\"
\n" ); document.write( "We just learned that tan squared minus sec squared is always equal to -1 as long as the two arguments are the same. So the expression in the parentheses will be -1:
\n" ); document.write( "\"6%28-1%29\"
\n" ); document.write( "which equals -6.

\n" ); document.write( "\"2sin%5E2+%28theta%2F4%29+%2B+2cos%5E2+%28theta%2F4%29\"
\n" ); document.write( "Here we factor out a 2:
\n" ); document.write( "\"2%28sin%5E2%28theta%2F4%29+%2B+cos%5E2%28theta%2F4%29%29\"
\n" ); document.write( "I hope by now you get that sin squared + cos squared is always equal to 1 as long as the two arguments are the same. So this expression becomes:
\n" ); document.write( "2(1)
\n" ); document.write( "or simply 2 \n" ); document.write( "
\n" );