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.
