Given that $\tan 3x = k \tan x$, with $k$ as a constant and $\tan x \neq 0$.
(i)[4]
After expanding $\tan(2x + x)$ first, show that $(3k - 1) \tan^2 x = k - 3$.
(ii)[3]
Hence solve $\tan 3x = k \tan x$ for $k = 4$, giving every solution in the interval $0^\circ < x < 180^\circ$.
(iii)[1]
Show that, when $k = 2$, the equation $\tan 3x = k \tan x$ has no root in the interval $0^\circ < x < 180^\circ$.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Apply the tan $(A+B)$ and tan $2A$ formulae to derive an equation in $\tan x$.” …