Example-1 :-  If sin x = 3/5, cos y = -12/13, where x and y both lie in second quadrant, find the value of sin (x + y).

  We know that sin (x + y) = sin x cos y + cos x sin y ... (1)
  cos² x = 1 - sin² x 
         = 1 - (3/5)²
         = 1 - 9/25
         = 16/25
  cos x = ± 4/5
  Here, x lies on the second quadrant, so cos x is negative.
  cos x = -4/5
  
  Now, sin² y = 1 - cos² y
         = 1 - (-12/13)²
         = 1 - 144/169
         = 25/169
  sin y = ± 5/13
  Here, y lies on the second quadrant, so sin y is positive.   
  sin y = 5/13

  So, put the put the values of sin x, siny, cos x and cos y in (1)
  sin (x + y) = sin x cos y + cos x sin y    
              = 3/5 x (-12/13) + (-4/5) x 5/13
              = -36/65 + (-20/65)
  sin (x + y) = -56/65       
    

Example-2 :-  Prove that: cos 2x . cos x/2 - cos 3x . cos 9x/2 = sin 5x . sin 5x/2

Example-3 :-  Find the value of tan π/8.

  Let x = π/8, then 2x = π/4
  Now, tan 2x = (2tan x)/(1 - tan² x)
       tan π/4 = (2tan π/8)/(1 - tan² π/8)
  Let y = tan π/8.
  Then, 1 = 2y/(1 - y²)
  1 - y² = 2y
  y² + 2y - 1 = 0
  By discrimination rule a = 1, b = 2, c = -1
  d = b² - 4ac
    = 2² - 4 x 1 x (-1)
    = 4 + 4
  d = 8
  Now, y = (-b ± √d)/2a
         = (-2 ± √8)/(2 x 1)
         = (-2 ± 2√2)/2
         = 2(-1 ± √2)/2
       y = -1 ± √2 
  Therefore, π/8 lies on the first quadrant, y = tan π/8 is positive.
  Hence, tan π/8 = -1 + √2 = √2 - 1
    

Example-4 :-  If tan x = 3/4, π < x < 3π/2, find the value of sin x/2, cos x/2 and tan x/2.

  Since π < x < 3π/2 (3rd quadrant), cos x is negative
  Also, π/2 < x/2 < 3π/4 (2nd quadrant)
  Therefore, sin x/2 is positive and cos x/2 is negative.
  Now,
  sec² x = 1 + tan² x 
         = 1 + (3/4)²
         = 1 + 9/16
         = 25/16
  sec x = 5/4
  1/cos x = 5/4
  cos x = 4/5 
  Here, x lies on the second quadrant, so cos x is negative.
  cos x = -4/5
  
  Now, 2sin² x/2 = 1 - cos x
                 = 1 - (-4/5)
                 = 1 + 4/5
                 = 9/5
        sin² x/2 = 9/10
         sin x/2 = 3/√10 (π/2 < x/2 < 3π/4, sin x/2 is positive)
 
  Now, cos² x/2 = 1 - sin² x/2
                = 1 - (3/√10)²
                = 1 - 9/10
       cos² x/2 = 1/10
        cos x/2 = 1/√10
        cos x/2 = -1/√10 (π/2 < x/2 < 3π/4, cos x/2 is negative)
 
  Now, tan x/2 = (sin x/2)/(cos x/2)
               = (3/√10)/(-1/√10)
               = 3/√10 x (-√10/1)
       tan x/2 = -3
    

Example-5 :-  Prove that: cos² x + cos² (x + π/3) + cos² (x - π/3) = 3/2