# Simplify the following?

## (1) ${\left(\sin \left(\frac{x}{2}\right) - \cos \left(\frac{x}{2}\right)\right)}^{2}$ (2) $\frac{{\cot}^{2} a - 1}{{\csc}^{2} a}$ (3) $\tan 2 a - \frac{1}{\cos 2 a}$

Mar 15, 2016

1)

expands to

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

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

2)

Expands to

$\frac{{\cot}^{2} a - 1}{{\csc}^{2} a} = \cos 2 a$

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

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

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

$\frac{\left(\sin a - \cos a\right) \left(\sin a - \cos a\right)}{\left(\sin a + \cos a\right) \left(\sin a - \cos a\right)}$

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

$\frac{\sin a - \cos a}{\sin a + \cos a}$

Divide numerator and denominator by $\sin a$

$\frac{1 - \cot a}{1 + \cot a}$