Write as a single term: cos (2x) cos (x) + sin (2x) sin (x)

Answer: cos (x)

Step-by-step explanation:

We have

sin (x + y) = sin (x) * cos (y) + cos (x) * sin (y) (1) and

cos (x + y) = cos (x) * cos (y) - sin (x) * sin (y) (2)

From eq. (1)

if x = y

sin (x + x) = sin (x) * cos (x) + cos (x) * sin (x) ⇒ sin (2x) = 2sin (x) cos (x)

From eq. 2

If x = y

cos (x + x) = cos (x) * cos (x) - sin (x) * sin (x) ⇒ cos² (x) - sin² (x)

cos (2x) = cos² (x) - sin² (x)

Hence:The expression:

cos (2x) cos (x) + sin (2x) sin (x) (3)

Subtition of sin (2x) and cos (2x) in eq. 3

[cos² (x) - sin² (x) ]*cos (x) + [ (2sen (x) cos (x) ]*sin (x)

and operating

cos³ (x) - sin² (x) cos (x) + 2sin² (x) cos (x) = cos³ (x) + sin² (x) cos (x)

cos (x) [ cos² (x) + sin² (x) ] = cos (x)

since cos² (x) + sin² (x) = 1