Question 6 - A-Level Maths

Show that $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) }$$ can be written as $\operatorname { cosec } ^ { 2 } x$.
Hence solve the equation $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) } = \operatorname { cosec } x + 3$$ giving the values of $x$ to the nearest degree in the interval $- 180 ^ { \circ } < x < 180 ^ { \circ }$.
Hence solve the equation $$\frac { \sec ^ { 2 } \left( 2 \theta - 60 ^ { \circ } \right) } { \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) + 1 \right) \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) - 1 \right) } = \operatorname { cosec } \left( 2 \theta - 60 ^ { \circ } \right) + 3$$ giving the values of $\theta$ to the nearest degree in the interval $0 ^ { \circ } < \theta < 90 ^ { \circ }$.