Trigonometry question: Simplify sin2u - cos2u to an expression using only sinu and cosu?

Question

You have to use trig identities such as sinu= 2sinucosu and cos2u=cos^2u-sin^2u

Answer 1

sin2u - cos2u
sin 2u = 2sinu*cosu
cos 2u = cos^2u - sin^2u = (cosu - sinu)(cosu + sinu)

So sin2u-cos2u= 2 sinu*cosu - (cosu-sinu)(cosu+sinu)

Answer 2

Without doing your problem for you look at the identities for sin2u and cos2u which reduce them to forms containing sinu and cosu. Then substitute and collect as many like terms as possible.

Although your books will not always tell you what use this is, it is foundational to finding and recognizing angular relationships. These are particularly useful in encoding frequencies to transmit and receive radio, TV, cell phone, etc. signals or create encryption formulas to restrict access to digital or analog information.

Answer 3

sin(2u) - cos(2u)
= sin(u+u) - cos(u+u)
= sin u cos u + cos u sin u - [ cos u cos u - sin u sin u ]
= 2 sin(u) cos(u) - cos^2(u) + sin^2(u)
= 2 sin(u) cos(u) - [1 - sin^2(u)] + sin^2(u)
= 2 sin(u) cos(u) - 1 + 2 sin^2(u)
= 2 sin(u) { cos(u) + sin(u) } - 1