Question

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

Answer 1

Answer: cos(x)

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