Just use the sum-to-product formulas...
sin(a)-sin(b) = 2cos((a+b)/2)sin((a-b)/2)
cos(a)-cos(b) = -2sin((a+b)/2)sin((a-b)/2)
So, let
a=8θ
b=10θ
and you have
sin(8θ)-sin(10θ) = 2cos(9θ)sin(-θ)
cos(10θ)-cos(8θ) = -2sin(9θ)sin(θ)
now it is clear to see that
2cos(9θ)sin(-θ) = cot(9θ)(-2sin(9θ)sin(θ))
The -2sin(θ) factors cancel, and you are left with
cos(9θ) = cot(9θ)sin(9θ)
which is true, since cot = cos/sin
Sin 8theta -sin 10theta= cot 9theta (cos 10theta - cos 8 theta)
3 answers
sin ( 8 θ ) - sin ( 10 θ ) = cot ( 9 θ ) [ cos ( 10 θ ) - cos ( 8 θ ) ]
sin ( 8 θ ) - sin ( 10 θ ) = [ cos ( 9 θ ) / sin ( 9 θ ) ] ∙ [ cos ( 10 θ ) - cos ( 8 θ ) ]
Multiply both sides by sin ( 9 θ )
sin ( 9 θ ) ∙ [ sin ( 8 θ ) - sin ( 10 θ ) ] = cos ( 9 θ ) ∙ [ cos ( 10 θ ) - cos ( 8 θ ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) - sin ( 9 θ ) ∙ sin ( 10 θ ) = cos ( 9 θ ) ∙ [ cos ( 10 θ ) - cos ( 8 θ ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) - sin ( 9 θ ) ∙ sin ( 10 θ ) = cos ( 9 θ ) ∙ cos ( 10 θ ) - cos ( 9 θ ) ∙ cos ( 8 θ )
_______________________________________
sin (A ) ∙ sin (B ) = ( 1 / 2 ) [ cos ( A - B ) - cos ( A + B ) ]
cos ( A ) ∙ cos ( B ) = ( 1 / 2 ) [ cos ( A - B ) + cos ( A + B ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 8 θ ) - cos ( 9 θ + 8 θ ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) = ( 1 / 2 ) [ cos ( θ ) - cos ( 17 θ ) ]
sin ( 9 θ ) ∙ sin ( 10 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 10 θ ) - cos ( 9 θ + 10 θ ) ]
sin ( 9 θ ) ∙ sin ( 10 θ ) = ( 1 / 2 ) [ cos ( - θ ) - cos ( 19 θ ) ]
Since:
cos ( - θ ) = cos ( θ )
sin ( 9 θ ) ∙ sin ( 10 θ ) = ( 1 / 2 ) [ cos ( θ ) - cos ( 19 θ ) ]
cos ( 9 θ ) ∙ cos ( 10 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 10 θ ) + cos ( 9 θ + 10 θ ) ]
cos ( 9 θ ) ∙ cos ( 10 θ ) = ( 1 / 2 ) [ cos ( - θ ) + cos ( 19 θ ) ]
Since:
cos ( - θ ) = cos ( θ
cos ( 9 θ ) ∙ cos ( 10 θ ) = ( 1 / 2 ) [ cos ( θ ) + cos ( 19 θ ) ]
cos ( 9 θ ) ∙ cos ( 8 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 8 θ ) + cos ( 9 θ + 8 θ ) ]
cos ( 9 θ ) ∙ cos ( 8 θ ) = ( 1 / 2 ) [ cos ( θ ) + cos ( 17 θ ) ]
_______________________________________
Replace this values in equation:
sin ( 9 θ ) ∙ sin ( 8 θ ) - sin ( 9 θ ) ∙ sin ( 10 θ ) = cos ( 9 θ ) ∙ cos ( 10 θ ) - cos ( 9 θ ) ∙ cos ( 8 θ )
( 1 / 2 ) [ cos ( θ ) - cos ( 17 θ ) ] - ( 1 / 2 ) [ cos ( θ ) - cos ( 19 θ ) ] = ( 1 / 2 ) [ cos ( θ ) + cos ( 19 θ ) ] - ( 1 / 2 ) [ cos ( θ ) + cos ( 17 θ ) ]
Multiply both sides by 2
cos ( θ ) - cos ( 17 θ ) - [ cos ( θ ) - cos ( 19 θ ) ] = cos ( θ ) + cos ( 19 θ ) - [ cos ( θ ) + cos ( 17 θ ) ]
cos ( θ ) - cos ( 17 θ ) - cos ( θ ) + cos ( 19 θ ) = cos ( θ ) + cos ( 19 θ ) - cos ( θ ) - cos ( 17 θ )
- cos ( 17 θ ) + cos ( 19 θ ) = cos ( 19 θ ) - cos ( 17 θ )
cos ( 19 θ ) - cos ( 17 θ ) = cos ( 19 θ ) - cos ( 17 θ )
This mean identity is true.
sin ( 8 θ ) - sin ( 10 θ ) = [ cos ( 9 θ ) / sin ( 9 θ ) ] ∙ [ cos ( 10 θ ) - cos ( 8 θ ) ]
Multiply both sides by sin ( 9 θ )
sin ( 9 θ ) ∙ [ sin ( 8 θ ) - sin ( 10 θ ) ] = cos ( 9 θ ) ∙ [ cos ( 10 θ ) - cos ( 8 θ ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) - sin ( 9 θ ) ∙ sin ( 10 θ ) = cos ( 9 θ ) ∙ [ cos ( 10 θ ) - cos ( 8 θ ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) - sin ( 9 θ ) ∙ sin ( 10 θ ) = cos ( 9 θ ) ∙ cos ( 10 θ ) - cos ( 9 θ ) ∙ cos ( 8 θ )
_______________________________________
sin (A ) ∙ sin (B ) = ( 1 / 2 ) [ cos ( A - B ) - cos ( A + B ) ]
cos ( A ) ∙ cos ( B ) = ( 1 / 2 ) [ cos ( A - B ) + cos ( A + B ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 8 θ ) - cos ( 9 θ + 8 θ ) ]
sin ( 9 θ ) ∙ sin ( 8 θ ) = ( 1 / 2 ) [ cos ( θ ) - cos ( 17 θ ) ]
sin ( 9 θ ) ∙ sin ( 10 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 10 θ ) - cos ( 9 θ + 10 θ ) ]
sin ( 9 θ ) ∙ sin ( 10 θ ) = ( 1 / 2 ) [ cos ( - θ ) - cos ( 19 θ ) ]
Since:
cos ( - θ ) = cos ( θ )
sin ( 9 θ ) ∙ sin ( 10 θ ) = ( 1 / 2 ) [ cos ( θ ) - cos ( 19 θ ) ]
cos ( 9 θ ) ∙ cos ( 10 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 10 θ ) + cos ( 9 θ + 10 θ ) ]
cos ( 9 θ ) ∙ cos ( 10 θ ) = ( 1 / 2 ) [ cos ( - θ ) + cos ( 19 θ ) ]
Since:
cos ( - θ ) = cos ( θ
cos ( 9 θ ) ∙ cos ( 10 θ ) = ( 1 / 2 ) [ cos ( θ ) + cos ( 19 θ ) ]
cos ( 9 θ ) ∙ cos ( 8 θ ) = ( 1 / 2 ) [ cos ( 9 θ - 8 θ ) + cos ( 9 θ + 8 θ ) ]
cos ( 9 θ ) ∙ cos ( 8 θ ) = ( 1 / 2 ) [ cos ( θ ) + cos ( 17 θ ) ]
_______________________________________
Replace this values in equation:
sin ( 9 θ ) ∙ sin ( 8 θ ) - sin ( 9 θ ) ∙ sin ( 10 θ ) = cos ( 9 θ ) ∙ cos ( 10 θ ) - cos ( 9 θ ) ∙ cos ( 8 θ )
( 1 / 2 ) [ cos ( θ ) - cos ( 17 θ ) ] - ( 1 / 2 ) [ cos ( θ ) - cos ( 19 θ ) ] = ( 1 / 2 ) [ cos ( θ ) + cos ( 19 θ ) ] - ( 1 / 2 ) [ cos ( θ ) + cos ( 17 θ ) ]
Multiply both sides by 2
cos ( θ ) - cos ( 17 θ ) - [ cos ( θ ) - cos ( 19 θ ) ] = cos ( θ ) + cos ( 19 θ ) - [ cos ( θ ) + cos ( 17 θ ) ]
cos ( θ ) - cos ( 17 θ ) - cos ( θ ) + cos ( 19 θ ) = cos ( θ ) + cos ( 19 θ ) - cos ( θ ) - cos ( 17 θ )
- cos ( 17 θ ) + cos ( 19 θ ) = cos ( 19 θ ) - cos ( 17 θ )
cos ( 19 θ ) - cos ( 17 θ ) = cos ( 19 θ ) - cos ( 17 θ )
This mean identity is true.
Thank you