How to Prove tan(a - b) = tana - tanb without using a prefix?

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To prove the identity tan(a - b) = (tan a - tan b) / (1 + tan a tan b), the discussion emphasizes using the sine and cosine definitions of tangent. The user attempts to substitute the sine and cosine formulas for a - b but feels stuck. Another participant suggests dividing both the numerator and denominator by cos a cos b to simplify the expression. This approach leads to the desired form of the tangent subtraction formula. The conversation highlights the importance of manipulating trigonometric identities to reach the proof.
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Homework Statement



Prove that: tan (a - b) = tana - tanb
1 + tanatanb​
]​

Homework Equations



cos (a- b) = cosacosb + sinasinb


sin (a - b) = sinacosb - cosasinb


The Attempt at a Solution



I have plugged in the two equations give above since tan - sin/cos and then I'm am stumped of how to proceed

tan (a - b) = sinacosb - cosasinb
cosacosb + sinasinb​

can anyone help please?
 
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Quinn Morris said:
tan (a - b) = sinacosb - cosasinb
cosacosb + sinasinb​

The formula you are trying to arrive at provides the clue: what could you divide numerator and denominator by to get that "1" in the denominator?
 
so times buy a form of one... 1 over cosacosb?
 
yes that is correct, divide the numerator and denominator by cosacosb
 

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