Simplify the following?

Question

(1) #(sin(x/2)-cos(x/2))^2#
(2) #(cot^2a-1)/ (csc^2a)#
(3) #tan2a-1/(cos2a)#

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Answer 1 · 2016-03-15T13:02:03

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)#