Show that cos x +cos(x+60)=3/5[1+root 3]
To prove that cos x + cos(x+60) = (3/5)[1+√3], we can use the trigonometric identity called the sum of two cosines:
cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)
Let's apply this identity to our equation.
cos x + cos(x+60) = 2cos((x + x + 60)/2)cos((x + x + 60)/2)
Simplifying,
cos x + cos(x+60) = 2cos((2x + 60)/2)cos((2x - 60)/2)
cos x + cos(x+60) = 2cos(x + 30)cos(x - 30)
Now, let's focus on simplifying cos(x + 30) and cos(x - 30).
Using the identity cos(A + B) = cos A cos B - sin A sin B,
cos(x + 30) = cos x cos 30 - sin x sin 30
= cos x(√3/2) - sin x(1/2)
= (√3/2)cos x - (1/2)sin x
Similarly,
cos(x - 30) = cos x cos (-30) - sin x sin (-30)
= cos x(√3/2) + sin x(1/2)
= (√3/2)cos x + (1/2)sin x
Substituting these back into our equation:
cos x + cos(x+60) = 2((√3/2)cos x - (1/2)sin x)((√3/2)cos x + (1/2)sin x)
= 2[(√3/2)cos^2 x - (1/2)sin x cos x + (1/2)sin x cos x + (1/2)sin^2 x]
= (√3/2)cos^2 x + (1/2)sin^2 x
Now, simplifying further using the identity cos^2 x + sin^2 x = 1:
cos x + cos(x+60) = (√3/2)cos^2 x + (1/2)sin^2 x
= (√3/2)cos^2 x + (1/2)(1 - cos^2 x)
= (√3/2)cos^2 x + 1/2 - (1/2)cos^2 x
= ((√3 - 1)/2)cos^2 x + 1/2
We can see that cos x + cos(x+60) is a function of cos^2 x with a coefficient of ((√3 - 1)/2).
Now, let's simplify the right-hand side of the given equation (3/5)[1+√3]:
(3/5)[1+√3] = (3/5) + (3/5)(√3)
Comparing both sides of the equation, we see that the left-hand side is a function of cos^2 x, while the right-hand side is a constant term that does not depend on x.
Hence, in order to prove that cos x + cos(x+60) = (3/5)[1+√3], we need to show that the left-hand side is equal to the right-hand side for all values of x.
To verify this equality, we can substitute various values of x and calculate the left-hand side and the right-hand side separately. If the values match for all these substitutions, it confirms that the two expressions are equal.
Alternatively, we can use a graphing calculator or software to plot the functions and verify that they coincide for all x.