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Video: Simplifying Trigonometric Expressions Using Double-Angle Identities

Alex Cutbill

Simplify sin 𝛼/(1 + tan 𝛼) โˆ’ sin 2𝛼/(2 cos 𝛼 + 2 sin 𝛼).

01:41

Video Transcript

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

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

Okay so now we have something in terms of only sin ๐›ผ and cos ๐›ผ. Letโ€™s simplify. We multiply the first fraction by cos ๐›ผ over cos ๐›ผ in an attempt to simplify the denominator. And of course because cos ๐›ผ over cos ๐›ผ is just one, this doesnโ€™t change the value of the fraction.

So now the first fraction is sin ๐›ผ cos ๐›ผ over cos ๐›ผ plus sin ๐›ผ. 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 ๐›ผ cos ๐›ผ over cos ๐›ผ plus sin ๐›ผ.

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