Cos9xcos7x - sin9xsin7x = - 1/2

Предыдущий
вопрос Следующий
вопрос

To simplify the left side of the equation, we can use the trigonometric identity for the difference of two angles:

cos(A)cos(B) - sin(A)sin(B) = cos(A + B)

Therefore, cos(9x)cos(7x) - sin(9x)sin(7x) = cos(9x + 7x) = cos(16x)

Now, we need to evaluate cos(16x):

cos(16x) = cos(90° - 74x)
= sin(74x)
= sin(60° + 14x)
= (√3/2)cos(14x) + (1/2)sin(14x)
= (√3/2)cos(14x) + (1/2)(√3/2)sin(14x)
= (√3/2)(cos(14x) + sin(14x))
= (√3/2)(cos(7x)(cos(7x) + sin(7x)) + sin(7x)(cos(7x) + sin(7x)))
= (√3/2)(cos(7x) + sin(7x))^2
= (√3/2)(cos^2(7x) + 2sin(7x)cos(7x) + sin^2(7x))
= (√3/2)(1 + 2sin(7x)cos(7x))
= (√3/2)(1 + sin(14x))
= (√3/2)(1 + sin(14x))

Therefore, the given equation becomes:
(√3/2)(1 + sin(14x)) = -1/2
1 + sin(14x) = -1/(√3)
sin(14x) = -1/(√3) - 1
sin(14x) = -√3/3 - 1
sin(14x) = -2√3/3

Therefore, the equation cos(9x)cos(7x) - sin(9x)sin(7x) = -1/2 is equivalent to sin(14x) = -2√3/3.