Prove Basic Problems (Dealing with Cos & Tan)

Question

Prove:

 

1-cos2x/1+cos2x=tan²x

 

Please include step by step and next to each step, please include the identity used! :)

Answer 1

Here's one way of going about it:

 

Remember the Double Angle Identity for Cosine:  cos2x = 2cos²x - 1

the Pythagorean Identity:  sin²x + cos²x = 1

and the tangent identity:  sin/cos = tan.

 

(1-cos2x)/(1+cos2x) = (1 - (2cos²x -1)) / (1 + (2cos²x - 1))    <Double Angle>

  =  (2 - 2cos²x) / 2cos²x      <simplify>

  =  (1 - cos²x) / cos²x         

  =  sin²x / cos²x                 <Pythagorean>

  =  tan²x                               <Tangent>