Recall that
• tan(x) = sin(x) / cos(x)
• sin(x) sin(y) = - 1/2 (cos(x + y) - cos(x - y))
• cos(x) cos(y) = 1/2 (cos(x + y) + cos(x - y))
Then we can write
tan(6x) tan(7x) = (sin(6x) sin(7x)) / (cos(6x) cos(7x))
tan(6x) tan(7x) = - (cos(13x) - cos(x)) / (cos(13x) + cos(x))
Make this replacement in the original equation and simplify:
tan(6x) tan(7x) = -1
- (cos(13x) - cos(x)) / (cos(13x) + cos(x)) = -1
(cos(13x) - cos(x)) / (cos(13x) + cos(x)) = 1
cos(13x) - cos(x) = cos(13x) + cos(x)
2 cos(x) = 0
cos(x) = 0
Solving for x gives
x = π/2 + nπ = (2n + 1)π/2
where n is any integer. However, since 2n + 1 is odd, for any odd multiple of π/2, we would end up with
tan(6 (2n + 1)π/2) = tan((2n + 1)×3π) = tan(odd multple of π/2)
since odd×odd = odd, and cos(odd multiplie of π/2) = 0, which would make tan(6x) undefined.
Therefore there are no solutions to this equation.