mathDemonstrate if sin(45+x) + sin(45-x) = 2^1/2*cosx.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll apply the sine function to the sum and the difference of angles 45 and x:

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

sin (a+b) = sin a*cos b + sin b*cos a

We'll put a = 45 and b = x.

sin (45+x) = sin 45*cos x + sin x*cos 45

sin (45+x) = sqrt2*cos x/2 + sqrt2*sin x/2 (1)

sin (45-x) = sin 45*cos x - sin x*cos 45

sin (45-x) = sqrt2*cos x/2 - sqrt2*sin x/2 (2)

Now, we'll substitute (1) and (2) in the given identity:

sin(45+x) + sin(45-x) = sqrt2*cosx

sqrt2*cos x/2 + sqrt2*sin x/2 + sqrt2*cos x/2 - sqrt2*sin x/2 = sqrt2*cosx

We'll combine and eliminate like terms:

2*sqrt2*cos x/2 = sqrt2*cosx

We'll simplify and we'll get:

sqrt2*cosx = sqrt2*cosx

The identity sin(45+x) + sin(45-x) = sqrt2*cosx is true.

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