Question

Which of the following expressions is equivalent to
(1 point)
csc x cos 2x - secx sin 2x
secxcos2x- csex sin 2x
secxcosx - cscx sin x
cscxcosx-secxsinx

Answer 1

The problem text looks garbled, so I'll simplify each of the four expressions you listed. Then you can tell me which one was supposed to be the target.

Let sinx ≠ 0 and cosx ≠ 0.

- csc x cos 2x - sec x sin 2x
= (cos2x)/sinx - (sin2x)/cosx
= (cos2x cosx - sin2x sinx)/(sinx cosx)
= cos(3x)/(sinx cosx).

- sec x cos 2x - csc x sin 2x
= (cos2x)/cosx - (sin2x)/sinx
= cos2x/cosx - 2 cosx
= (cos2x - 2cos^2 x)/cosx
= -1/cosx
= -sec x.

- sec x cos x - csc x sin x
= (1/cosx)cosx - (1/sinx)sinx
= 1 - 1
= 0.

- csc x cos x - sec x sin x
= cosx/sinx - sinx/cosx
= (cos^2 x - sin^2 x)/(sinx cosx)
= cos2x/(sinx cosx).

So:
- csc x cos 2x - sec x sin 2x = cos3x/(sinx cosx).
- None of the other three expressions simplifies to cos3x/(sinx cosx) for all x (they simplify to -sec x, 0, and cos2x/(sinx cosx), respectively).

If you tell me exactly which expression was being matched (or paste the original unformatted problem), I’ll pick the correct choice.