6. (i) Use an appropriate double angle formula to show that cosec2x = λ cosec x sec x, and state the value of the constant λ - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 8

To solve the equation, we start with:

$3sec²θ + 3secθ - 2tan²θ = 0$

We know that:

$tan²θ = sec²θ - 1$ Substituting this into the equation gives:

$3sec²θ + 3secθ - 2(sec²θ - 1) = 0$ This simplifies to:

$3sec²θ + 3secθ - 2sec²θ + 2 = 0$ Combining like terms leads to:

$sec²θ + 3secθ + 2 = 0$ Now, let’s factor the quadratic equation:

$(secθ + 1)(secθ + 2) = 0$ From this, we have two possible solutions:

1. $secθ + 1 = 0 \implies \secθ = -1$
2. $secθ + 2 = 0 \implies \secθ = -2$

Next, we convert the secant values back to cosine:

1. $secθ = -1 \implies \cosθ = -1 \implies θ = \pi$
2. $secθ = -2 \implies \cosθ = -\frac{1}{2} \implies θ = \frac{2\pi}{3}, \frac{4\pi}{3}$

Thus, the complete solution set for the range 0 < θ < 2π is:

$θ = \pi, \frac{2\pi}{3}, \frac{4\pi}{3}.$