Cosx + cos2x + cos6x + cos7x= 4cos x/2 cos 5x/2 cos4x

Предыдущий
вопрос Следующий
вопрос

To simplify this expression, we will use the sum-to-product formula for cosine:

cos(A) + cos(B) = 2 cos[(A + B) / 2] cos[(A - B) / 2]

Now, let's apply this formula to the given expression:

cosx + cos2x + cos6x + cos7x
= cos(x) + cos(2x) + cos(6x) + cos(7x)
= 2 cos[(x + 7x) / 2] cos[(x - 7x) / 2] + 2 cos[(2x + 6x) / 2] cos[(2x - 6x) / 2]
= 2 cos(4x) cos(-3x) + 2 cos(4x) cos(-2x)
= 2 cos(4x) cos(3x) + 2 cos(4x) cos(2x)
= 2 cos(4x) (cos(3x) + cos(2x))

Therefore, the simplified expression is:

2 cos(4x) (cos(3x) + cos(2x)
= 2 cos(x/2) cos(5x/2) * cos(4x)