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Find the value of the trignometric expression \mathrm{tan\left ( arctan\ 4x-arctan\ 3 \right )+tan\left ( arctan\ 4-arctan\ 3x \right )=\frac{1}{2}.}

Option: 1

\frac{1}{10}


Option: 2

\frac{1}{2}


Option: 3

\frac{1}{5}


Option: 4

\frac{1}{4}


Answers (1)

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Option (a) \frac{1}{10}

Given that,

\mathrm{tan\left ( arctan\ 4x-arctan\ 3 \right )+tan\left ( arctan\ 4-arctan\ 3x \right )=\frac{1}{2} }

Using formula

\mathrm{ \frac{tan\ A+tan\ B}{1-tan\ A\ tan\ B}=tan\ \theta }

\mathrm{\frac{4x-3}{1+\left ( 4x \right )\left ( 3 \right )}+\frac{4-3x}{1+\left ( 4 \right )\left ( 3x \right )}=\frac{1}{2} }

\mathrm{\frac{4x-3}{1+12x}+\frac{4-3x}{1+12x}=\frac{1}{2} }

\mathrm{ \frac{4x-3+4-3x}{1+12x}=\frac{1}{2} }

\mathrm{\frac{x+1}{1+12x}=\frac{1}{2} }

\mathrm{2x+2=1+12x }

\mathrm{x=\frac{1}{10} }

Posted by

SANGALDEEP SINGH

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