Trigonometry
      Trigonometric functions of double angles, double angle formulas
         Trigonometric functions expressed by the half angle
         Trigonometric functions of double angles expressed by the tangent function
         Trigonometric functions expressed by the tangent of the half angle
      Half angle formulas
      Trigonometric functions expressed by the cosine of the double angle
      Trigonometric identities, examples
Trigonometric functions of double angles, double angle formulas
By substituting b with a in the sum formulas,
sin (a + b) = sin a · cos b + cosa · sin b      and     cos (a + b) = cos a · cos b - sin a · sin b,
  and
thus for example,   sin 2a = sin (a + a) = sin a · cos a + cosa · sin a = 2sin a cos a   so we get,
  sin 2a = 2 sin a cos a and cos 2a = cos2 a - sin2 a  
and
The double angle formula for the cosine function can be expressed by sine or cosine function using the
identity sin2 a + cos2 a = 1,
  cos 2a = 2cos2 a - 1 or cos 2a = 1 - 2sin2 a  
and  1 + cos 2a = 2 cos2 a or 1 - cos 2a = 2sin2 a  
Trigonometric functions expressed by the half angle
Substituting a/2 in the double angle formulas we obtain trigonometric functions expressed by the half angle,
and
and
and
or
or
Trigonometric functions of double angles expressed by the tangent function
The double angle formulas can be expressed in terms of a single function, so using the identity
and dividing the numerator and denominator by cos2 a we get  
and from  
dividing the right side by cos2 a gives  
  then    
Trigonometric functions expressed by the tangent of the half angle
Replacing a by a/2 in the above identities, we get
         
Half angle formulas
Using the identities in which trigonometric functions are expressed by the half angle,
=>   =>  
and applying the definitions of the functions, tangent and cotangent 
=>   =>  
Trigonometric functions expressed by the cosine of the double angle
Replacing a/2 by a in the above identities, we get
         
Trigonometric identities examples
Example:   Using known values, sin 60° = Ö3/2 and sin 45° = Ö2/2 evaluate sin 105°.
Solution:  Applying the sum formula for the sine function,  sin (a + b) = sin a · cos b + cosa · sin b, 
therefore,   sin 105° = sin (60° + 45°) = sin 60° · cos 45° + cos60° · sin 45°
Example:   Use,  tan 45° = 1 and  tan 60° = Ö3,  to prove that  tan 15° = 2 - Ö3.
Solution:
Example:   Prove the identity
Solution:   Using the addition formula
Example:   Verify the identity  
Solution:   We divide the numerator and denominator on the left side by sin a and to the right side we use 
the cotangent formula for the difference of two angles, thus
Example:   Express sin 3x in terms of sin x.
Solution:   Using the sum formula and the double angle formula for the sine function,
sin 3x = sin (2x + x) = sin 2x · cos x + cos 2x · sin x = 2sin x cos x · cos x + (cos2 x - sin2 x) · sin x
            = 2sin x · (1 - sin2 x) + (1 - 2sin2 x) · sin x = 3sin x - 4sin3 x.
Example:   Express tan 3x in terms of tan x.
Solution:   Using the sum formula and the double angle formula for the tangent function,
Example:   Prove the identity  
Solution:
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