Video Transcript
Which one of the following is an equivalent expression to seven over the csc of 𝑥 plus 12 degrees? Is it (A) seven cos of 𝑥 plus 12 degrees? (B) Seven cos of 𝑥 minus 12 degrees. (C) Seven cos of 78 degrees minus 𝑥. (D) Seven sin of 78 degrees minus 𝑥. Or (E) sin of 78 degrees minus 𝑥 divided by seven.
In order to simplify our expression, we begin by recalling one of the reciprocal trigonometric identities. This states that one over sin 𝜃 is equal to csc 𝜃. And as a result, one over csc 𝜃 is equal to sin 𝜃. The argument in our expression is 𝑥 plus 12 degrees. So we can therefore rewrite seven over csc of 𝑥 plus 12 degrees as seven sin of 𝑥 plus 12 degrees.
This does not match any of the given five answer options. So we will need to simplify our expression further. To do this, we recall one of the cofunction identities. The cos of 90 degrees minus 𝜃 is equal to sin 𝜃. Once again, we will substitute the argument in this question 𝑥 plus 12 degrees for 𝜃. Seven sin of 𝑥 plus 12 degrees is therefore equal to seven cos of 90 degrees minus 𝑥 plus 12 degrees. Distributing negative one across the parenthesis on the right-hand side, we have seven cos of 90 degrees minus 𝑥 minus 12 degrees. And since 90 minus 12 is 78, this in turn simplifies to seven cos of 78 degrees minus 𝑥.
The expression seven over the csc of 𝑥 plus 12 degrees is equivalent to seven cos of 78 degrees minus 𝑥, which was option (C).
