### 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.