Question

Log tan 17°+ log tan 37°+log tan 53°+ log tan 73°=

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\log\,tan\,17^\circ+\log\,tan\,37^\circ+\log\,tan\,53^\circ+\log\,tan\,73^\circ}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}$

$\mathsf{\log\,tan\,17^\circ+\log\,tan\,37^\circ+\log\,tan\,53^\circ+\log\,tan\,73^\circ}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Product rule:}}$

$\mathsf{\log(M_1\,M_2\,M_3\,M_4)=\log\,M_1+\log\,M_2+\log\,M_3+\log\,M_4}$

$\mathsf{Consider,}$

$\mathsf{\log\,tan\,17^\circ+\log\,tan\,37^\circ+\log\,tan\,53^\circ+\log\,tan\,73^\circ}$

$\textsf{using product rule of logarithm, we get}$

$\mathsf{=\log(tan\,17^\circ{\times}tan\,37^\circ{\times}tan\,53^\circ{\times}tan\,73^\circ)}$

$\mathsf{But}\;\;\boxed{\mathsf{tan\theta=cot(90^\circ-\theta)}}$

$\mathsf{=\log(tan\,17^\circ{\times}tan\,37^\circ{\times}cot(90^\circ-53^\circ){\times}cot(90^\circ-73^\circ)\;)}$

$\mathsf{=\log(tan\,17^\circ{\times}\,tan\,37^\circ{\times}\\,cot\,37^\circ{\times}\,cot\,17^\circ\;)}$

$\textsf{Using}\;\boxed{\mathsf{cot\,\theta=\dfrac{1}{tan\theta}}}$

$\mathsf{=\log(tan\,17^\circ{\times}tan\,37^\circ{\times}\dfrac{1}{tan\,37^\circ}{\times}\dfrac{1}{tan\,17^\circ}\;)}$

$\mathsf{=\log(1)}$

$\mathsf{=0}$

$\implies\boxed{\mathsf{\log\,tan\,17^\circ+\log\,tan\,37^\circ+\log\,tan\,53^\circ+\log\,tan\,73^\circ=0}}$