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 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 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, leaving us with one minus cos squared 𝜃. Next, we recall one of the
Pythagorean identities: sin squared 𝜃 plus cos squared 𝜃 is equal to one. Subtracting cos squared 𝜃 from
both sides, this can be rewritten as sin squared 𝜃 is equal to one minus cos
squared 𝜃. This means that our expression can
be rewritten as sin squared 𝜃. sin 𝜃 multiplied by csc 𝜃 minus
cos squared 𝜃 written in its simplest form is sin squared 𝜃.
