6. a. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$( \cos x + \sin x ) ( \operatorname { cosec } x - \sec x ) \equiv 2 \cot 2 x$$ b. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { 2 } \sin \left( 2 x - \frac { \pi } { 6 } \right)$$