Express 1 - 2*cos(2a) as a product
One way:
1 - 2*cos(2x)
= sin^2 x + cos^2 x - 2(cos^2 x - sin^2 x)
= 3sin^2 x - cos^2 x
= (√3sinx + cosx)(√3sinx - cosx)
oops, replace all x's with a's, makes no difference
To express the expression 1 - 2*cos(2a) as a product, we can use the trigonometric identity for the cosine of twice an angle:
cos(2a) = 1 - 2*sin^2(a)
Replacing cos(2a) in the expression 1 - 2*cos(2a), we obtain:
1 - 2*(1 - 2*sin^2(a))
Now, let's simplify:
1 - 2 + 4*sin^2(a)
Combining like terms:
-1 + 4*sin^2(a)
By factoring out a common factor of 4, we have:
4*(sin^2(a) - 1/4)
Now, we can use the difference of squares formula:
sin^2(a) - 1/4 = (sin(a) - 1/2)(sin(a) + 1/2)
Putting it all together, we get:
1 - 2*cos(2a) = -1 + 4*sin^2(a) = 4*(sin(a) - 1/2)(sin(a) + 1/2)