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
explanation: The document contains multiple math questions along with their solutions. Specifically, it includes two questions related to trigonometric functions, each with multiple parts. The questions involve sketching graphs, proving trigonometric identities, and finding the number of solutions to equations by drawing suitable lines on the same axes.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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distilabel_metadata.role: system
distilabel_metadata.content: # 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains multiple math questions along with their solutions. Specifically, it includes two questions related to trigonometric functions, each with multiple parts. The questions involve sketching graphs, proving trigonometric identities, and finding the number of solutions to equations by drawing suitable lines on the same axes.
  
  Label: multiple-question-and-answer