Standard +0.8 Part (a) requires proving an identity involving reciprocal trig functions and double angles, demanding algebraic manipulation of cot and tan into cos/sin form. Part (b) applies this result to solve an equation with compound angles, requiring substitution, double angle application, and careful angle arithmetic within constraints. This goes beyond routine C3/C4 exercises, requiring multi-step reasoning and comfort with reciprocal functions, but remains accessible with standard techniques.
12. (a) Show that
$$\cot x - \tan x \equiv 2 \cot 2 x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
(b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\)
$$5 + \cot \left( \theta - 15 ^ { \circ } \right) - \tan \left( \theta - 15 ^ { \circ } \right) = 0$$
giving your answers to one decimal place. [0pt]
[Solutions based entirely on graphical or numerical methods are not acceptable.]
12. (a) Show that
$$\cot x - \tan x \equiv 2 \cot 2 x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
(b) Hence, or otherwise, solve, for $0 \leqslant \theta < 180 ^ { \circ }$
$$5 + \cot \left( \theta - 15 ^ { \circ } \right) - \tan \left( \theta - 15 ^ { \circ } \right) = 0$$
giving your answers to one decimal place.\\[0pt]
[Solutions based entirely on graphical or numerical methods are not acceptable.]
\hfill \mbox{\textit{Edexcel C34 2018 Q12 [9]}}