(1-cos(x))(1+cos(x)= 1/(csc^2(x)) verify the identity?

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Answer 1 · 2018-04-04T22:18:11

Distribute the left hand side and use the Pythagorean Theorem to conclude that #(1-cosx)(1+cosx)=1/csc^2x#.

Given

#(1-cosx)(1+cosx)#

Apply distributive property (or FOIL if you like)

#(1-cosx)(1+cosx)=1-cos^2x#

Apply the Pythagorean Theorem

#1-cos^2x=sin^2x#

The definition of #cscx# says that #cscx=1/sinx#, so #sinx=1/cscx#, so we can square both sides to get

#sin^2x=1/csc^2x#.

Therefore

#(1-cosx)(1+cosx)=1/csc^2x#