text: # 6.8.1 Trigonometric Functions Long Questions (Question 1 & 2)
  
  Question 1:
  (a) Sketch the graph of y = cos 2x for 0°  x  180°.
  (b) Hence, by drawing a suitable straight line on the same axes, find the number of solutions satisfying the equation for 0°  x  180°.
  
  Solution:
  
  (a)(b)
  
  Question 2:
  (a) Prove that $\frac{2\mathrm{tan}x}{2-{\mathrm{sec}}^{2}x}=\mathrm{tan}2x.$
  (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 $\frac{3x}{\pi }+\frac{2\mathrm{tan}x}{2-{\mathrm{sec}}^{2}x}=0$  for 0  x π .
  State the number of solutions.
  
  Solution:
  (a)
  $\begin{array}{l}\frac{2\mathrm{tan}x}{2-{\mathrm{sec}}^{2}x}=\mathrm{tan}2x\\ \\ LHS:\\ \frac{2\mathrm{tan}x}{2-{\mathrm{sec}}^{2}x}=\frac{2\mathrm{tan}x}{2-\left(1+{\mathrm{tan}}^{2}x\right)}\\ \text{}=\frac{2\mathrm{tan}x}{2-{\mathrm{tan}}^{2}x}\\ \text{}=\mathrm{tan}2x\text{(RHS)}\end{array}$
  
  (b)(i)
  
  (b)(ii)
  
  When x = 0, y = 0.
  When x = π, = 3.
  Number of solutions = 3