Q) the value of tan to the power of negative 1 end exponent open parentheses fraction numerator a subscript 1 x minus y over denominator a subscript 1 y plus x end fraction close parentheses space plus space tan to the power of negative 1 end exponent open parentheses fraction numerator a subscript 2 minus a subscript 1 over denominator 1 plus a subscript 1 a subscript 2 end fraction close parentheses space plus space tan to the power of negative 1 end exponent open parentheses fraction numerator a subscript 3 minus a subscript 2 over denominator 1 plus a subscript 2 a subscript 3 end fraction close parentheses plus.... plus space tan to the power of negative 1 end exponent open parentheses fraction numerator a subscript n minus a subscript n minus 1 end subscript over denominator 1 plus a subscript n a subscript n minus 1 end subscript end fraction close parentheses plus space tan to the power of negative 1 end exponent 1 over a subscript n
w h e r e space x comma y comma a subscript 1 comma a subscript 2 comma... comma a subscript n element of R to the power of plus comma i s space e q u a l space t o
left parenthesis A right parenthesis tan to the power of negative 1 end exponent open parentheses y over x close parentheses space space left parenthesis B right parenthesis tan to the power of negative 1 end exponent open parentheses y over x close parentheses minus straight pi over 4 space space space space left parenthesis C right parenthesis tan to the power of negative 1 end exponent open parentheses x over 4 close parentheses minus straight pi over 4 space space space left parenthesis D right parenthesis tan to the power of negative 1 end exponent open parentheses x over y close parentheses

Asked by araima2001 | 5th Apr, 2017, 05:28: AM

Expert Answer:

begin mathsize 16px style tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript 1 straight x minus straight y over denominator straight a subscript 1 straight y plus straight x end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript 2 minus straight a subscript 1 over denominator 1 plus straight a subscript 1 straight a subscript 2 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript 3 minus straight a subscript 2 over denominator 1 plus straight a subscript 3 straight a subscript 2 end fraction close parentheses plus........... plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript straight n minus an minus 1 over denominator 1 plus straight a subscript straight n straight a subscript straight n minus 1 end subscript end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over straight a subscript straight n close parentheses
Dividing space by space straight x space to space numerator space and space denominator space only space for space the space first space term comma
tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript 1 minus begin display style straight y over straight x end style over denominator straight a subscript 1 begin display style straight y over straight x end style plus 1 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript 2 minus straight a subscript 1 over denominator 1 plus straight a subscript 1 straight a subscript 2 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript 3 minus straight a subscript 2 over denominator 1 plus straight a subscript 3 straight a subscript 2 end fraction close parentheses plus........... plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript straight n minus an minus 1 over denominator 1 plus straight a subscript straight n straight a subscript straight n minus 1 end subscript end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over straight a subscript straight n close parentheses
tan to the power of negative 1 end exponent straight a subscript 1 minus tan to the power of negative 1 end exponent straight y over straight x plus tan to the power of negative 1 end exponent straight a subscript 2 minus tan to the power of negative 1 end exponent straight a subscript 1 plus tan to the power of negative 1 end exponent straight a subscript 3 minus tan to the power of negative 1 end exponent straight a subscript 2 plus....... plus tan to the power of negative 1 end exponent straight a subscript straight n minus tan to the power of negative 1 end exponent straight a subscript straight n minus 1 end subscript plus tan to the power of negative 1 end exponent open parentheses 1 over straight a subscript straight n close parentheses
After space cancelling space the space terms space we space get comma
minus tan to the power of negative 1 end exponent straight y over straight x plus tan to the power of negative 1 end exponent straight a subscript straight n plus tan to the power of negative 1 end exponent open parentheses 1 over straight a subscript straight n close parentheses
minus tan to the power of negative 1 end exponent straight y over straight x plus straight pi over 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets Using space tan to the power of negative 1 end exponent straight x plus space tan to the power of negative 1 end exponent straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus straight y over denominator 1 minus straight x end fraction close parentheses close square brackets
straight pi over 2 minus tan to the power of negative 1 end exponent straight y over straight x
cot to the power of negative 1 end exponent fraction numerator straight y space over denominator straight x end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets using space tan to the power of negative 1 end exponent straight x plus space cot to the power of negative 1 end exponent straight x equals straight pi over 2 close square brackets space space space space space space space space space space space space space space space space space space space space space space
tan to the power of negative 1 end exponent straight x over straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets Using space space cot to the power of negative 1 end exponent straight x equals tan to the power of negative 1 end exponent 1 over straight x close square brackets
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Answered by Sneha shidid | 5th Apr, 2017, 10:27: AM