Question

given that cos(45+x)cos(45-x) =1/2cos2x prove the above identity
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Answer 1

Answer:

sin(45+x)·sin(45-x) = (1/2)cos2x

Use the sum and difference of two angles identity:

(sin45cosx+sinxcos45)·(sin45cosx-sinxcos45) = (1/2)cos2x

This is of the form (a+b)(a-b) = a2 - b2, so it's:

sin245cos2x - sin2xcos245 = (1/2)cos2x

Use The Pythagorean Identity sin2θ+cos2θ=1, so sin2θ=1-cos2θ. Also sin245 = cos245 = 1/2

(1/2)cos2x - (1-cos2x)(1/2) = (1/2)cos2x

(1/2)(cos2x - 1 + cos2x) = (1/2)cos2x

(1/2)(2cos2x - 1) = (1/2)cos2x

Last identity: cos2x = 2cos2x - 1:

(1/2)cos2x = (1/2)cos2x