3. (a) Use the identity for \(\cos ( A + B )\) to prove that \(\cos 2 A = 2 \cos ^ { 2 } A - 1\).
(b) Use the substitution \(x = 2 \sqrt { } 2 \sin \theta\) to prove that $$\int _ { 2 } ^ { \sqrt { 6 } } \sqrt { \left( 8 - x ^ { 2 } \right) } \mathrm { d } x = \frac { 1 } { 3 } ( \pi + 3 \sqrt { } 3 - 6 ) .$$ A curve is given by the parametric equations $$x = \sec \theta , \quad y = \ln ( 1 + \cos 2 \theta ) , \quad 0 \leq \theta < \frac { \pi } { 2 } .$$ (c) Find an equation of the tangent to the curve at the point where \(\theta = \frac { \pi } { 3 }\).