Question

find the value of tan18+tan42+root3tan18×tan42

Answer 1

$\textbf{To find:}$

$\textsf{The value of}$

$\mathsf{tan18^\circ+tan42^\circ+\sqrt{3}\,tan18^\circ\,tan42^\circ}$

$\textbf{Solution:}$

$\mathsf{Consider,}$

$\mathsf{tan60^\circ=tan(18^\circ+42^\circ)}$

$\mathsf{Using,}$

$\boxed{\mathsf{tan(A+B)=\dfrac{tanA+tanB}{1-tanA\,tanB}}}$

$\mathsf{tan60^\circ=\dfrac{tan18^\circ+tan42^\circ}{1-tan18^\circ\,tan42^\circ}}$

$\mathsf{\sqrt{3}=\dfrac{tan18^\circ+tan42^\circ}{1-tan18^\circ\,tan42^\circ}}$

$\mathsf{tan18^\circ+tan42^\circ=\sqrt{3}(1-tan18^\circ\,tan42^\circ)}$

$\mathsf{tan18^\circ+tan42^\circ=\sqrt{3}-\sqrt{3}\,tan18^\circ\,tan42^\circ)}$

$\implies\boxed{\mathsf{tan18^\circ+tan42^\circ+\sqrt{3}\,tan18^\circ\,tan42^\circ)=\sqrt{3}}}$

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