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))^= (√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)) = -1/1 + sin(14x) = -1/(√3sin(14x) = -1/(√3) - sin(14x) = -√3/3 - 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.
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))^
= (√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/
1 + sin(14x) = -1/(√3
sin(14x) = -1/(√3) -
sin(14x) = -√3/3 -
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.