# What is the sum of cos(30-x)+cos(30+x)?

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We'll apply the following identities:

cos (a-b) = cosa*cos b + sin b*sin a

cos (a+b) = cosa*cos b - sin b*sin a

We'll put a = 30 and b = x.

cos (30+x) = cos 30*cos x - sin x*sin 30

cos (30+x) = sqrt3*cos x/2 - sin x/2 (1)

cos (30-x) = cos 30*cos x + sin x*sin30

cos (30-x) = sqrt3*cos x/2 + sin x/2 (2)

Now, we'll replace (1) and (2) in the sum to be calculated:

cos (30+x) + cos (30-x) = sqrt3*cos x/2 - sin x/2 + sqrt3*cos x/2 + sin x/2

We'll combine and eliminate like terms:

cos (30+x) + cos (30-x) = 2*sqrt3*cos x/2

We'll simplify and we'll get:

**cos (30+x) + cos (30-x) = sqrt3*cosx**