Home Forums Trigonometry If $$tanA = \frac{sinB}{1-cosB}$$ find tan2A

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    If $tanA = \frac{sinB}{1-cosB}$ find $tan2A$

    #24272
    admin
    Keymaster

    To find ( \tan(2A) ) given that ( \tan(A) = \frac{\sin(B)}{1 – \cos(B)} ), we’ll first express ( \tan(A) ) in a more familiar trigonometric identity and then use the double angle formula for ( \tan ).

    Given:
    
    \[ \tan(A) = \frac{\sin(B)}{1 – \cos(B)} \]
    
    Recall the trigonometric identity:
    
    \[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \]
    
    However, our expression looks similar but not exactly the same. Notice that for \( \frac{\sin(B)}{1 – \cos(B)} \), if we consider \( \theta = 2B \), we get:
    
    \[ \tan\left(\frac{2B}{2}\right) = \frac{\sin(2B)}{1 + \cos(2B)} \]
    
    But our denominator is \(1 – \cos(B)\), not \(1 + \cos(B)\). To relate this to the given equation, let’s use the identity:
    
    \[ \tan(A) = \frac{1 – \cos(\theta)}{\sin(\theta)} \]
    
    This is a form of the half-angle identity. However, to directly find \( \tan(2A) \) from \( \tan(A) = \frac{\sin(B)}{1 – \cos(B)} \), let’s directly apply the double angle formula for tangent, knowing that \( \tan(A) \) is given. The double angle formula is:
    
    \[ \tan(2A) = \frac{2\tan(A)}{1 – \tan^2(A)} \]
    
    Substituting \( \tan(A) = \frac{\sin(B)}{1 – \cos(B)} \) into the formula:
    
    \[ \tan(2A) = \frac{2\left(\frac{\sin(B)}{1 – \cos(B)}\right)}{1 – \left(\frac{\sin(B)}{1 – \cos(B)}\right)^2} \]
    
    Let’s simplify this expression.
    
    After simplifying the expression, we find that:
    
    \[ \tan(2A) = -\tan(B) \]
    
    This result shows that \( \tan(2A) \) is directly related to \( -\tan(B) \), given the initial relationship between \( \tan(A) \) and \( \sin(B) \) and \( \cos(B) \).
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