Answer on Question #51811 – Math – Trigonometry
Deduce the identity of :
1−2sin2A
A sin 2A
B tan 2A
C cos 2A
D cos A
Solution
Method 1
This method uses ideas of Trigonometry.
Recall formulas
cos 2 𝐴 + sin2 𝐴 = 1
and cos(2𝐴) = cos 2 𝐴 − sin2 𝐴, which follows from
cos(𝐴 + 𝐵) = cos𝐴 ∙ cos𝐵 − sin𝐴 ∙ sin𝐵.
Then
1 − 2sin2 𝐴 = (cos 2 𝐴 + sin2 𝐴) − 2sin2 𝐴 =
= cos 2 𝐴 + (sin2 𝐴 − 2sin2 𝐴) = cos 2 𝐴 − sin2 𝐴 = cos(2𝐴)
Method 2
This method uses ideas of Complex Analysis.
According to Euler's formula sin    ei  ei  / 2 ,
 exp i   exp  i  
 exp  2i   2 exp i  exp  i   exp  2i  
1  2sin   1  2  
  1 2
2i
4i 2


2
2
 1
 exp  2i   exp  2i   cos 2
1
exp  2i   2  exp  2i  

2
2
Answer: C cos (2A)
www.AssignmentExpert.com