Question Video: Simplifying Trigonometric Expressions Using Pythagorean and Reciprocal Identities Mathematics • 10th Grade

Simplify sin πœƒ csc πœƒ βˆ’ cosΒ² πœƒ.

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Video Transcript

Simplify sin πœƒ multiplied by csc πœƒ minus cos squared πœƒ.

In this question, we are asked to simplify a trigonometric expression. One way of doing this is using the reciprocal and Pythagorean identities. In questions of this type, it is not always clear what to do first. However, as a general rule, it is worth replacing any reciprocal functions with the sine, cosine, or tangent function.

We know that csc πœƒ is equal to one over sin πœƒ. Substituting this into our expression, we have sin πœƒ multiplied by one over sin πœƒ minus cos squared πœƒ. The sin πœƒ on the numerator and denominator of our first term cancels, leaving us with one minus cos squared πœƒ. Next, we recall one of the Pythagorean identities: sin squared πœƒ plus cos squared πœƒ is equal to one. Subtracting cos squared πœƒ from both sides, this can be rewritten as sin squared πœƒ is equal to one minus cos squared πœƒ. This means that our expression can be rewritten as sin squared πœƒ. sin πœƒ multiplied by csc πœƒ minus cos squared πœƒ written in its simplest form is sin squared πœƒ.

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