Standard +0.3 Part (a) requires algebraic manipulation of the addition formula using the given constraint, which is straightforward once you substitute tan α = 2 cot β and simplify. Part (b) is a standard trigonometric equation that converts to a quadratic in tan θ using sec²θ = 1 + tan²θ, then requires finding angles in the given range. Both parts use routine A-level techniques with no novel insight required, making this slightly easier than average.
a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that
$$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$
b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation
$$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$
a) Given that $\alpha$ and $\beta$ are two angles such that $\tan \alpha = 2 \cot \beta$, show that
$$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$
b) Find all values of $\theta$ in the range $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$ satisfying the equation
$$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$
\hfill \mbox{\textit{WJEC Unit 3 2019 Q9}}