Video: Simplifying Trigonometric Expressions Using Reciprocal Identities

Simplify cosΒ² πœƒ sec πœƒ csc πœƒ.

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Video Transcript

Simplify cos squared πœƒ times sec πœƒ times csc πœƒ.

Using some trig identities, we can replace a few of these things and maybe have some things cancel and then we can simplify. So we know that the sec πœƒ is equal to one over cos πœƒ, and then csc πœƒ is equal to one over sin πœƒ. So by replacing those, the cosine on the denominator will cancel with one of the cosines on the numerator. Since it’s cosine squared, that’s the same as having two cosines.

So on the numerator, we have cosine times one times one, so we have cos πœƒ, and then on the bottom, there’s only a sin of πœƒ. Now this does simplify a little bit more. The reason why is because cosine divided by sine has a relationship because if we would flip that sin of πœƒ divided by cos of πœƒ, that is equal to tan of πœƒ; it simplifies.

Now we could write that as tan of πœƒ over one, so that means that if we would flip that upside down, it would be equal to what we had: cos of πœƒ divided by sin of πœƒ. So we have one divided by the tan of πœƒ, and we actually know what that is. One divided by the tan of πœƒ is equal to the cot of πœƒ. So after simplifying, our final answer would be cot of πœƒ.

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