SOLUTION: Use pythagorgean identities to write the expression as an integer. tan^2 6beta - sec^2 6beta 6 tan^2 beta - 6 sec^2 beta 2sin^2 (theta/4) + 2cos^2 (theta/4)

Question 595159: Use pythagorgean identities to write the expression as an integer.
tan^2 6beta - sec^2 6beta
6 tan^2 beta - 6 sec^2 beta
2sin^2 (theta/4) + 2cos^2 (theta/4)
Answer by jsmallt9(3758) (Show Source): 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

we are learning that sin squared of anything + cos squared of that same thing add up to a 1:

etc.

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

Once we get this "place-holder" idea, problems like yours become very easy.

The Pythagorean identity that involves tan and sec is:

With a little algebra we can work this around to look like your expression. Subtracting 1 and from each side we get:

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!

Here we start by factoring out a 6:

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:

which equals -6.

Here we factor out a 2:

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:
2(1)
or simply 2
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