# How do you simplify 2cos(10x) + 4cos(5x)?

Sep 18, 2015

Simplify y = 2cos (10x) + 4cos (5x)

Ans: (cos 5x - 1 - sqrt3)(cos 5x + 1 - sqrt3)

#### Explanation:

Replace 2cos (10x) by $4 {\cos}^{2} \left(5 x\right) - 2$, we get:
y = 4cos^2 (5x) + 4cos (5x) - 2
y = 2cos^2 (5x) + 2cos (5x) - 1.
D = d^2 = b^2 - 4ac = 4 + 8 = 12 --> d = +- 2sqrt3
2 real roots:
cos 5x = 1 + sqrt3 and cos 5x = - 1 + sqrt3
Factored form or simplified form :
y = (cos 5x - 1 - sqrt3)(cos 5x + 1 - sqrt3)