Question 9 - A-Level Maths

Exam Board	OCR
Module	C3 (Core Mathematics 3)
Year	2007
Session	June
Topic	Addition & Double Angle Formulae

Prove the identity $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta - 3 } { 1 - 3 \tan ^ { 2 } \theta }$$
Solve, for $0 ^ { \circ } < \theta < 180 ^ { \circ }$, the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = 4 \sec ^ { 2 } \theta - 3 ,$$ giving your answers correct to the nearest $0.1 ^ { \circ }$.
Show that, for all values of the constant k , the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = \mathrm { K } ^ { 2 }$$ has two roots in the interval $0 ^ { \circ } < \theta < 180 ^ { \circ }$.