Final answer:
To solve the equation 2sin(x/2) sin(x) = sqrt2 sin(x), factor out sin(x) and solve for both sin(x) = 0 and 2sin(x/2) - sqrt2 = 0. The solutions are x = 0, x = pi, x = 90 degrees + 4npi, or x = 270 degrees + 4npi.
Step-by-step explanation:
To solve the equation 2sin(x/2) sin(x) = sqrt2 sin(x) using a combination of factoring and trigonometric identities, we can start by factoring out sin(x) on the left side:
2sin(x/2) sin(x) = sqrt2 sin(x)
sin(x) (2sin(x/2) - sqrt2) = 0
Now, we have two factors: sin(x) = 0 or 2sin(x/2) - sqrt2 = 0
For sin(x) = 0, the solutions are x = 0 and x = pi.
For 2sin(x/2) - sqrt2 = 0, we can rewrite it as sin(x/2) = sqrt2/2 and use the trigonometric identity sin(45 degrees) = sqrt2/2:
x/2 = 45 degrees + 2npi or x/2 = 135 degrees + 2npi
x = 90 degrees + 4npi or x = 270 degrees + 4npi
Therefore, the solutions to the equation 2sin(x/2) sin(x) = sqrt2 sin(x) are x = 0, x = pi, x = 90 degrees + 4npi, or x = 270 degrees + 4npi, where n is an integer.