SOLUTION: Use tan2θ= sin2θ/cos2θ to prove the double angle formula for tangent: tan2θ= 2tanθ/1-tan^2(theta) can someone please help me with this equation and explain

Algebra ->  Trigonometry-basics -> SOLUTION: Use tan2θ= sin2θ/cos2θ to prove the double angle formula for tangent: tan2θ= 2tanθ/1-tan^2(theta) can someone please help me with this equation and explain      Log On


   

Question 980781: Use tan2θ= sin2θ/cos2θ to prove the double angle formula for tangent: tan2θ= 2tanθ/1-tan^2(theta) can someone please help me with this equation and explain it to me? Thank you.
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!

See the lesson  Addition and subtraction formulas  in this site,
section  "Proof of the addition and subtraction formulas for tangents".

Or,  below is a direct derivation.

We have

     tan%282theta%29 = sin%282theta%29%2Fcos%282theta%29,                         (1)

according to the definition of tangents as the ratio of sines and cosines.

Next,

  sin%282theta%29 = 2%2Asin%28theta%29%2Acos%28theta%29.                 (2)

This is the formula for sines of the double argument.  Similarly,

  cos%282theta%29 = cos%5E2%28theta%29+-+sin%5E2%28theta%29.             (3)

This is the formula for cosines of the double argument.

Now,  substitute  (2)  and  (3)  into the numerator and the denominator of  (1).  You will get

tan%282theta%29 = %282%2Asin%28theta%29%2Acos%28theta%29%29%2F%28cos%5E2%28theta%29+-+sin%5E2%28theta%29%29.             (4)

Now,  divide the numerator and denominator in the right side of  (4)  by  cos%5E2%28theta%29.  You will get

tan%282theta%29 = .                     (5)

As a last step,  replace the ratio  sin%28theta%29%2Fcos%28theta%29  by  tan%28theta%29  in  (5).  You will get

tan%282theta%29 = 2%2Atan%28theta%29%2F%281+-+tan%5E2%28theta%29%29.                     (6)

This is what has to be proved.  You finally got the formula you needed.