Cos(60°+x)cosx+sin(60°+x)sinx=0.5

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To simplify the equation, we can use the trigonometric identity:

cos(A + B) = cosAcosB - sinAsinB

Substitute A = 60° and B = x into the identity:

cos(60° + x) = cos60°cosx - sin60°sinx

Similarly, we have:

sin(60° + x) = sin60°cosx + cos60°sinx

Now, we can substitute these expressions into the given equation:

(cos60°cosx - sin60°sinx)cosx + (sin60°cosx + cos60°sinx)sinx = 0.5

Expanding the equation:

cos60°cos^2(x) - sin60°sin(x)cos(x) + sin60°cos(x)sin(x) + cos60°sin^2(x) = 0.5

cos60°cos^2(x) + cos60°sin^2(x) = 0.5

cos60° = 0.5

Therefore, the equation simplifies to:

0.5 + 0.5 = 0.5

This shows that the equation is true for all values of x.