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Video: Simplifying Trigonometric Expressions Using Double-Angle Identities

Alex Cutbill

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

01:41

Video Transcript

Simplify sin 𝛼 over one plus tan 𝛼 minus sin two 𝛼 over two cos 𝛼 plus two sin 𝛼.

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

Okay so now we have something in terms of only sin 𝛼 and cos 𝛼. Let’s simplify. We multiply the first fraction by cos 𝛼 over cos 𝛼 in an attempt to simplify the denominator. And of course because cos 𝛼 over cos 𝛼 is just one, this doesn’t change the value of the fraction.

So now the first fraction is sin 𝛼 cos 𝛼 over cos 𝛼 plus sin 𝛼. 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 𝛼 cos 𝛼 over cos 𝛼 plus sin 𝛼.

We can notice at least two terms of the same, and so when we subtract one from the other, we get zero. So sin 𝛼 over one plus tan 𝛼 minus sin two 𝛼 over two cos 𝛼 plus two sin 𝛼 is simply equal to zero, and you can’t get much simpler than that.