1)
expands to
sin^2 (x/2) + cos^2(x/2) - 2sin (x/2)cos(x/2)
= 1 - 2sin (x/2)cos(x/2) = 1 - sin x
2)
Expands to
(cot^2a-1)/ (csc^2a) = cos2a
(cos^2a/sin^2a-1)/ (csc^2a) = ((cos^2a - sin^2a)/sin^2a)/csc^2a
((cos^2a - sin^2a)/sin^2a)/(1/sin^2a) = cos^2a - sin^2a = cos 2a
3)
tan 2a -1/ (cos2a
After bringing to common denominator and simplifying
(sin 2a)/(cos2a) -1/( cos2a
(sin 2a -1)/( cos2a
(2sinacosa -sin^2a - cos ^2 a)/(cos^2 a - sin^ 2a
Multiple both numerator and denominator by -1
(-2sinacosa +sin^2a + cos ^2 a)/(sin^ 2a - cos^2 a
((sina - cosa)(sin a - cos a))/((sina + cosa) (sina - cos a))
(cancel((sina - cosa))(sin a - cos a))/((sina + cosa) cancel((sina - cos a))
(sina - cosa)/(sina + cosa)
Divide numerator and denominator by sina
(1 - cota)/(1 + cota)