How do you simplify cos(u+v)cosv+sin(u+v)sinvcos(u+v)cosv+sin(u+v)sinv?

1 Answer
Noah G
Sep 5, 2016

The expression can be simplified to cosucosu.

Explanation:

We need to start by expanding the cos(A + B)cos(A+B) and the sin(A + B)sin(A+B) using the sum and difference identities, as shown in the following image.

![http://www.regentsprep.org/regents/math/algtrig/att14/http://formulalesson.htm](https://useruploads.socratic.org/5YltLONvT86Xt1YrQQBu_formul30.gif)

Expanding, we have:

=>(cosucosv - sinusinv)(cosv) + (sinucosv + cosusinv)(sinv)⇒(cosucosv−sinusinv)(cosv)+(sinucosv+cosusinv)(sinv)

=>cosucos^2v - sinusinvcosv + sinucosvsinv + cosusin^2v⇒cosucos2v−sinusinvcosv+sinucosvsinv+cosusin2v

=>cosucos^2v + cosusin^2v⇒cosucos2v+cosusin2v

=>cosu(cos^2v + sin^2v)⇒cosu(cos2v+sin2v)

Applying the pythagorean identity sin^2x + cos^2x = 1sin2x+cos2x=1:

=>cosu⇒cosu

Hopefully this helps!

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