Prove or disprove the following identity:
cos (x+pi/6) +sin (x-pi/3) = 0
I will use 30° and 60° , easier to type
cos (x+300) +sin (x-60)
= cosxcos30 - sinxsin30 + sinxcos60 - cosxsin60
= (√3/2)cosx - (1/2)sinx + (1/2)sinx - (√3/2)cosx
= 0
= RS
To prove or disprove the given identity, we need to show whether it holds true for all values of x or not. Let's simplify both sides of the equation and see if they are equal.
Using the sum and difference identities for cosine and sine, we can rewrite the equation:
cos(x)cos(pi/6) - sin(x)sin(pi/6) + sin(x)cos(pi/3) - cos(x)sin(pi/3) = 0
Next, let's simplify the trigonometric functions with the known values:
(cos(x) * sqrt(3)/2) - (sin(x) * 1/2) + (sin(x) * 1/2) - (cos(x) * sqrt(3)/2) = 0
Combine the like terms:
(cos(x) - cos(x)) * sqrt(3)/2 + (sin(x) - sin(x)) * 1/2 = 0
Since the common factors cancel out, we are left with:
0 = 0
We can see that the equation holds true for all values of x. Thus, the given identity is proven to be true.