Simplify the following?

Question

Simplify the following?

(1) ${(sin (\frac{x}{2}) - cos (\frac{x}{2}))}^{2}$ $(sin(x/2)-cos(x/2))^2$
(2) $\frac{{cot}^{2} a - 1}{{csc}^{2} a}$ $(cot^2a-1)/ (csc^2a)$
(3) $tan 2 a - \frac{1}{cos 2 a}$ $tan2a-1/(cos2a)$

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Answer 1 · 2016-03-15T13:02:03

1)

expands to

${sin}^{2} (\frac{x}{2}) + {cos}^{2} (\frac{x}{2}) - 2 sin (\frac{x}{2}) cos (\frac{x}{2})$ $sin^2 (x/2) + cos^2(x/2) - 2sin (x/2)cos(x/2)$

$= 1 - 2 sin (\frac{x}{2}) cos (\frac{x}{2}) = 1 - sin x$ $= 1 - 2sin (x/2)cos(x/2) = 1 - sin x$

2)

Expands to

$\frac{{cot}^{2} a - 1}{{csc}^{2} a} = cos 2 a$ $(cot^2a-1)/ (csc^2a) = cos2a$

$\frac{\frac{{cos}^{2} a}{{sin}^{2} a} - 1}{{csc}^{2} a} = \frac{\frac{{cos}^{2} a - {sin}^{2} a}{{sin}^{2} a}}{{csc}^{2} a}$ $(cos^2a/sin^2a-1)/ (csc^2a) = ((cos^2a - sin^2a)/sin^2a)/csc^2a$

$\frac{\frac{{cos}^{2} a - {sin}^{2} a}{{sin}^{2} a}}{\frac{1}{{sin}^{2} a}} = {cos}^{2} a - {sin}^{2} a = cos 2 a$ $((cos^2a - sin^2a)/sin^2a)/(1/sin^2a) = cos^2a - sin^2a = cos 2a$

3)

$tan 2 a - \frac{1}{cos 2 a}$ $tan 2a -1/ (cos2a$

After bringing to common denominator and simplifying

$\frac{sin 2 a}{cos 2 a} - \frac{1}{cos 2 a}$ $(sin 2a)/(cos2a) -1/( cos2a$

$\frac{sin 2 a - 1}{cos 2 a}$ $(sin 2a -1)/( cos2a$

$\frac{2 sin a cos a - {sin}^{2} a - {cos}^{2} a}{{cos}^{2} a - {sin}^{2} a}$ $(2sinacosa -sin^2a - cos ^2 a)/(cos^2 a - sin^ 2a$

Multiple both numerator and denominator by -1

$\frac{- 2 sin a cos a + {sin}^{2} a + {cos}^{2} a}{{sin}^{2} a - {cos}^{2} a}$ $(-2sinacosa +sin^2a + cos ^2 a)/(sin^ 2a - cos^2 a$

$\frac{(sin a - cos a) (sin a - cos a)}{(sin a + cos a) (sin a - cos a)}$ $((sina - cosa)(sin a - cos a))/((sina + cosa) (sina - cos a))$

$(cancel((sina - cosa))(sin a - cos a))/((sina + cosa) cancel((sina - cos a))$

$(sina - cosa)/(sina + cosa)$

Divide numerator and denominator by $sina$

$(1 - cota)/(1 + cota)$

Science

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Simplify the following?

(1) ${(sin (\frac{x}{2}) - cos (\frac{x}{2}))}^{2}$ $(sin(x/2)-cos(x/2))^2$
(2) $\frac{{cot}^{2} a - 1}{{csc}^{2} a}$ $(cot^2a-1)/ (csc^2a)$
(3) $tan 2 a - \frac{1}{cos 2 a}$ $tan2a-1/(cos2a)$

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Simplify the following?

(1) (sin(x/2)-cos(x/2))^2(sin(x2)−cos(x2))2(sin(x/2)-cos(x/2))^2 (2) (cot^2a-1)/ (csc^2a)cot2a−1csc2a(cot^2a-1)/ (csc^2a) (3) tan2a-1/(cos2a)tan2a−1cos2atan2a-1/(cos2a)

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Impact of this question

(1) ${(sin (\frac{x}{2}) - cos (\frac{x}{2}))}^{2}$ $(sin(x/2)-cos(x/2))^2$
(2) $\frac{{cot}^{2} a - 1}{{csc}^{2} a}$ $(cot^2a-1)/ (csc^2a)$
(3) $tan 2 a - \frac{1}{cos 2 a}$ $tan2a-1/(cos2a)$