Long Question 3

Question 3:

(a) Prove that

\frac{2 \tan x}{2 - \sec^{2} x} = \tan 2 x .

(b)(i) Sketch the graph of y = – tan 2x for 0 ≤ x ≤ π .

(b)(ii) Hence, by drawing a suitable straight line on the same axes, find the number of solutions satisfying the equation

\frac{3 x}{π} + \frac{2 \tan x}{2 - \sec^{2} x} = 0

for 0 ≤ x ≤ π .

State the number of solutions.

Solution:

(a)

\begin{array}{l} \frac{2 \tan x}{2 - \sec^{2} x} = \tan 2 x \\ L H S : \\ \frac{2 \tan x}{2 - \sec^{2} x} = \frac{2 \tan x}{2 - (1 + \tan^{2} x)} \\ = \frac{2 \tan x}{2 - \tan^{2} x} \\ = \tan 2 x (RHS) \end{array}

(b)(i)

(b)(ii)

\begin{array}{l} \frac{3 x}{π} + \frac{2 \tan x}{2 - \sec^{2} x} = 0 \\ \frac{3 x}{π} + \tan 2 x = 0 \leftarrow from part (a) \\ - \tan 2 x = \frac{3 x}{π} \\ y = \frac{3 x}{π} \\ The suitable straight line to sketch is y = \frac{3 x}{π} . \end{array}

When x = 0, y = 0.

When x = π, y = 3.

Number of solutions = 3