Simplify the following?

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

1 Answer
Mar 15, 2016

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)