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Hi, my name is Jake, and I am one of the Math 130 SI leaders. Today we're going to talk
about how to simplify trigonometric identities. Ok, so we're going to verify this identity
right here. We have 1 over 1 minus sine x minus 1 over 1 plus sine x equals 2 times
secant x tangent x. So when you're verifying identities like this,
usually what you want to do is start with one side of the equation and work to get the
other one. Usually you want to pick the more difficult
side and work until you get the easier side. So I'm going to pick the left side of this
one because it has fractions and things like that; it looks a little more complicated.
So I'm just rewriting what we have on the left side of the equals sign, and our end
goal is to get 2 times secant x tangent x. So another guideline you can use for proving
these identities is to write what you have on this side of the equals sign in terms of
sine and cosine. So this already has sines in it, so we're
good, we don't have to rewrite it in terms of sine and cosine, it already is.
So the next guideline you can use is that if you have fractions like this, you need
to combine them to get one fraction because here we have two fractions, but here it's
just one term. So if we combined these fractions here, to
get the numerator, we do 1 times 1 plus sine x, minus 1 times 1 minus sine x.
Ok, so we just found a common denominator. And then the common denominator is the product
of these two, so 1 minus sine x times 1 plus sine x.
Ok, so if we simplify both the numerator and the denominator, the numerator should go to
2 times sine x. And then the denominator, you can see this
is just factored as a difference of squares, so it's 1 minus sine squared x.
So this was combine like terms, and this one was FOIL.
So we can then use an identity in the denominator, so 1 minus sine squared x can be written as
cosine squared x. That comes from this identity here [written].
So this one's probably the most popular Pythagorean identity, sin^2(x) + cos^2(x) = 1.
So if you solved for cosine squared, you'd get 1 minus sine squared.
So this is where we get the cosine squared now.
So now what we want to do is we can rewrite this fraction, as long as we have terms that
are being multiplied together, we can kind of split these things apart as we need them.
So we can rewrite this as 2 times let's have 1 over cosine x and then sine x over cosine
x. So you can see this is still the same thing
because in the numerator, we have 2 times 1 times sine x, which is 2sin(x).
In the denominator, we have cosine x times cosine x, and that gives you cosine squared,
so we're just splitting things apart. Ok, now you can see we're really close to
this right here. We can use our reciprocal identities now, so we can say this is 2, 1
over cosine x is secant x; sine x over cosine x is tangent x.
So this comes from our reciprocal identities. So this does it. This shows that this side
of the equals sign is equal to the right side of the equals sign.
This has been my explanation for simplifying trig identities.