Question Video: Simplifying Trigonometric Expressions Using Pythagorean and Reciprocal Identities Mathematics

Simplify sin 𝜃csc 𝜃 − cos² 𝜃.

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Video Transcript

Simplify sin 𝜃 multiplied by csc 𝜃 minus cos squared 𝜃.

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 we should do first. However, as a general rule, it is worth starting by replacing any reciprocal functions with the sine, cosine, and tangent functions. We know that csc 𝜃 is equal to one over sin 𝜃. Substituting this into our expression, we have sin 𝜃 multiplied by one over sin 𝜃 minus cos squared 𝜃. The sin 𝜃 on the numerator and denominator of our first term cancels. So we are left with one minus cos squared 𝜃.

Next, we recall one of our Pythagorean identities. sin squared 𝜃 plus cos squared 𝜃 is equal to one. Subtracting cos squared 𝜃 from both sides, we have sin squared 𝜃 is equal to one minus cos squared 𝜃. This means that our expression simplifies to sin squared 𝜃. sin 𝜃 multiplied by csc 𝜃 minus cos squared 𝜃 in its simplest form is sin squared 𝜃.

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