Trigonometry
      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 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:   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:
Example:   Prove the identity  
Solution:
Example:   If  tan a = 3/4, find tan a/2.   
Solution:   Use formula  to express tan a/2  in terms of  tan a.
Example:   Prove the identity  
Solution:   Substitute  then
Example:   Prove the identity  
Solution:   Using the formula for the sum of the tangent 
Example:   Prove that  
Solution:  Replace sin a by cos (p/2 - a) and cos a by sin (p/2 - a) and use the sum to product formula
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