Using these known identities:
#color(white){color(black)(
(sin(-theta)=-sintheta, qquad(1)),
(sin^2theta+cos^2theta=1, qquad(2.1)),
(cos^2theta=1-sin^2theta, qquad(2.2) ) :}#
(Identity #(2.2)# is achieved by subtracting #sin^2theta# from both sides of identity #(2.1)#.)
We can prove the identity (I will be manipulating the left side of the equation until it equals the right side):
#color(white){color(black)(
((1+siny)(1+sin(-y)), qquad"Left hand side"),
((1+siny)(1-siny), qquad"Use identity "(1)),
(1-siny+siny-sin^2y, qquad"Use FOIL method"),
(1color(red)cancel(color(black)(-siny+siny))-sin^2y, qquad"Cancel like terms"),
(1-sin^2y, qquad"Rewrite"),
(cos^2y, qquad"Use identity " (2.2)):}#
That's it! We just proved the identity because the left side equals the right side.
