How do you verify #(1 + cos theta)(1 - cos theta) = sin^2 theta#?

2 Answers
May 25, 2015

Use Pythagoras and the basic definitions of sine and cos
sin x = #o/h#
cos x = #a/h#
where o =the side opposite the angle x, a = the side adjacent to the angle x and h equals the hypotenuse of the right-angled triangle.

Pythagora states #h^2 = a^2 + o^2#
therefore #1 = a^2/h^2 + o^2/h^2#
so 1 = #cos^2(x) + sin^2 (x)#
#sin^2(x) = 1 - cos^2(x)#
tthen #sin^2(x) = (1 - cos(x))(1+cos(x))#

May 26, 2015

#(1+cos theta)(1-cos theta) = 1-cos^2 theta#

# = (sin^2 theta + cos^2 theta )- cos^2 theta#

# = sin^2 theta + (cos^2 theta - cos^2 theta)#

# = sin^2 theta#