LET US PROVE Prove the following identities when they are defined: 1. \( \cos ^{2} x-\cos ^{2} y=\sin ^{2} y-\sin ^{2} \chi \) 2. \( \sin ^{4} x-\cos ^{4} x=(\sin x-\cos x)(\sin x+\cos x) \) 3. \( \frac{1-\cos x}{\sin x}-\frac{\sin x}{1+\cos x}=0 \) (4.) \( \frac{1}{1+\tan ^{2} x}-\frac{1}{1+\tan ^{2} y}=\sin ^{2} y-\sin ^{2} x \)

Find the quotient. Leave the result in trigonometric form. (Le \[ \frac{\cos (\pi)+i \sin (\pi)}{\cos (2 \pi / 3)+i \sin (2 \pi / 3)} \]

\( \cos ( \alpha - \beta ) = \cos ( \alpha ) - \cos ( \beta ) ( \pi ) \)

\( \sqrt { 3 } \operatorname { ctg } ( x + \frac { \pi } { 3 } ) \geq - 1 \)

Find the product. Leave the result in trigonometric form. (Enter your an \[ \left[\frac{1}{3}\left(\cos \left(140^{\circ}\right)+i \sin \left(140^{\circ}\right)\right)\right]\left[\frac{4}{5}\left(\cos \left(240^{\circ}\right)+i \sin \left(240^{\circ}\right)\right)\right] \]

Q. Calcular el ángulo de los siguientes númeios complejos a) \( 3+2 i \) b) \( \frac{5}{2}+\frac{3}{7} i \)