### Video Transcript

Simplify sin ๐ผ over one plus tan
๐ผ minus sin two ๐ผ over two cos ๐ผ plus two sin ๐ผ.

So Iโve copies down the expression
to simplify, and my goal first of all is to get everything in terms of sin ๐ผ and
cos ๐ผ. That means rewriting tan ๐ผ as sin
๐ผ over cos ๐ผ and sin two ๐ผ as two times sin ๐ผ times cos ๐ผ, where here we used
the double angle identity for sine.

Okay so now we have something in
terms of only sin ๐ผ and cos ๐ผ. Letโs simplify. We multiply the first fraction by
cos ๐ผ over cos ๐ผ in an attempt to simplify the denominator. And of course because cos ๐ผ over
cos ๐ผ is just one, this doesnโt change the value of the fraction.

So now the first fraction is sin ๐ผ
cos ๐ผ over cos ๐ผ plus sin ๐ผ. Is there anything we can do to
simplify the second fraction before we perform the subtraction? Yes, the numerator and denominator
have a common factor of two, which we can cancel out. So we are left with sin ๐ผ cos ๐ผ
over cos ๐ผ plus sin ๐ผ.

We can notice at least two terms of
the same, and so when we subtract one from the other, we get zero. So sin ๐ผ over one plus tan ๐ผ
minus sin two ๐ผ over two cos ๐ผ plus two sin ๐ผ is simply equal to zero, and you
canโt get much simpler than that.