Question

if 2tanA+cotA=tanB then cotA+2tan(A-B)=

Answer 1

Answer on Question #49551 – Math – Trigonometry

If $2 \tan A + \cot A = \tan B$ then $\cot A + 2 \tan (A - B) = ?$

Solution.

$$\begin{array}{l} \cot A + 2 \tan (A - B) = \frac {1}{\tan A} + 2 \frac {\sin (A - B)}{\cos (A - B)} = \frac {1}{\tan A} + 2 \frac {\sin A \cos B - \sin B \cos A}{\cos A \cos B + \sin A \sin B} = \\ = \frac {1}{\tan A} + 2 \frac {(\sin A \cos B - \sin B \cos A) : (\sin A \sin B)}{(\cos A \cos B + \sin A \sin B) : (\sin A \sin B)} = \frac {1}{\tan A} + 2 \frac {\cot B - \cot A}{\cot A \cot B + 1} = \\ = \frac {\cot A \cot B + 1 + 2 \tan A (\cot B - \cot A)}{\tan A (\cot A \cot B + 1)} = \frac {\cot B (2 \tan A + \cot A) - 2 \tan A \cot A + 1}{\tan A (\cot A \cot B + 1)} = \\ = \frac {\cot B \tan B - 1}{\tan A (\cot A \cot B + 1)} = \frac {1 - 1}{\tan A (\cot A \cot B + 1)} = 0. \end{array}$$

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