Question

Prove the following identity:

(sec x -1)^2 - (tan x - sin x)^2 = (1 - cos x)^2

Answer 1

sec=1/cos
tan=sin/cos
so,....by using (a-b)²=a²-2ab+b²,
we get,
(1/c²)-(2/c)+1-(s²/c²)+2s²/c-(s²)
i have bolded,some terms in specific manner....we will work on them...
by pythagoras' identity,
sin²+cos² = 1
so....we wll get,
=2+2cos-sin²
=1+1+2cos-sin²
=cos²-2cos+1
=(1-cos)²...........proved