Ques:

Asked by sourabh7jain | 16th Jun, 2016, 05:20: PM

Expert Answer:

begin mathsize 12px style tan open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator straight d 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 d over denominator 1 plus straight a subscript 2 straight a subscript 3 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight d over denominator 1 plus straight a subscript 3 straight a subscript 4 end fraction close parentheses plus... plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight d over denominator 1 plus straight a subscript straight n minus 1 end subscript straight a subscript straight n end fraction close parentheses close square brackets
Replacing space apostrophe straight d apostrophe space by space straight a subscript 2 minus straight a subscript 1 comma space straight a subscript 3 minus straight a subscript 2 comma space straight a subscript 4 minus straight a subscript 3 comma.... straight a subscript straight n minus straight a subscript straight n minus 1 end subscript space in space different space tan to the power of negative 1 end exponent space terms comma space we space get

tan open square brackets 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 2 straight a subscript 3 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight a subscript 4 minus straight a subscript 3 over denominator 1 plus straight a subscript 3 straight a subscript 4 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 straight a subscript straight n minus 1 end subscript over denominator 1 plus straight a subscript straight n minus 1 end subscript straight a subscript straight n end fraction close parentheses close square brackets
equals tan open square brackets open parentheses 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 close parentheses plus open parentheses 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 close parentheses plus open parentheses tan to the power of negative 1 end exponent straight a subscript 4 minus tan to the power of negative 1 end exponent straight a subscript 3 close parentheses plus... plus open parentheses 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 close parentheses close square brackets
equals tan open square brackets 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 1 close square brackets space left parenthesis all space other space terms space get space cancelled right parenthesis
equals fraction numerator tan open parentheses tan to the power of negative 1 end exponent straight a subscript straight n close parentheses minus tan open parentheses tan to the power of negative 1 end exponent straight a subscript 1 close parentheses over denominator 1 plus tan open parentheses tan to the power of negative 1 end exponent straight a subscript straight n close parentheses minus tan open parentheses tan to the power of negative 1 end exponent straight a subscript 1 close parentheses end fraction
equals fraction numerator straight a subscript straight n minus straight a subscript 1 over denominator 1 plus straight a subscript straight n straight a subscript 1 end fraction end style

Answered by satyajit samal | 17th Jun, 2016, 12:17: PM