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
(