How do you prove #sin (x+y)sin(x-y) = cos^2y-cos^2x#?
1 Answer
Jan 22, 2016
You should use the following trigonometrical identities:
#sin (x + y) = sin x cos y + cos x sin y #
#sin (x - y) = sin x cos y - cos x sin y #
This will lead you to:
#sin(x + y )sin (x - y)#
#= (sin x cos y + cos x sin y)(sin x cos y - cos x sin y)#
# = (color(blue)(sin x cos y) + color(red)(cos x sin y))(color(blue)(sin x cos y) - color(red)(cos x sin y))#
... use the formula
# = sin^2 x cos^2 y - cos^2 x sin^2 y#
... use
# = (1 - cos^2 x) cos^2 y - cos^2 x (1 - cos^2 y)#
# = cos^2 y - cos^2 x cos^2 y - cos^2 x + cos^2 x cos^2 y#
# = cos^2 y - cancel(cos^2 x cos^2 y) - cos^2 x + cancel(cos^2 x cos^2 y)#
# = cos^2 y - cos^2 x#