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)#
